networks using a novel metaheuristic algorithm

energies

Article

Optimal Power Flow for Transmission PowerNetworks Using a Novel Metaheuristic Algorithm

Zelan Li 1, Yijia Cao 1,*, Le Van Dai 2 , Xiaoliang Yang 1,3 and Thang Trung Nguyen 4,*1 College of Electrical and Information Engineering, Hunan University, Changsha 410082, China;

[email protected] (Z.L.); [email protected] (X.Y.)2 Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam;

[email protected] College of Electrical and Information Engineering, Zhengzhou University of Light Industry,

Zhengzhou 450002, China4 Power System Optimization Research Group, Faculty of Electrical and Electronics Engineering,

Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam* Correspondence: [email protected] (Y.C.); [email protected] (T.T.N.)

Received: 14 October 2019; Accepted: 7 November 2019; Published: 12 November 2019��

Abstract: In the paper, a modified coyote optimization algorithm (MCOA) is proposed for findinghighly effective solutions for the optimal power flow (OPF) problem. In the OPF problem, totalactive power losses in all transmission lines and total electric generation cost of all available thermalunits are considered to be reduced as much as possible meanwhile all constraints of transmissionpower systems such as generation and voltage limits of generators, generation limits of capacitors,secondary voltage limits of transformers, and limit of transmission lines are required to be exactlysatisfied. MCOA is an improved version of the original coyote optimization algorithm (OCOA) withtwo modifications in two new solution generation techniques and one modification in the solutionexchange technique. As compared to OCOA, the proposed MCOA has high contributions as follows:(i) finding more promising optimal solutions with a faster manner, (ii) shortening computationsteps, and (iii) reaching higher success rate. Three IEEE transmission power networks are used forcomparing MCOA with OCOA and other existing conventional methods, improved versions of theseconventional methods, and hybrid methods. About the constraint handling ability, the success rate ofMCOA is, respectively, 100%, 96%, and 52% meanwhile those of OCOA is, respectively, 88%, 74%,and 16%. About the obtained solutions, the improvement level of MCOA over OCOA can be up to30.21% whereas the improvement level over other existing methods is up to 43.88%. Furthermore,these two methods are also executed for determining the best location of a photovoltaic system (PVS)with rated power of 2.0 MW in an IEEE 30-bus system. As a result, MCOA can reduce fuel cost andpower loss by 0.5% and 24.36%. Therefore, MCOA can be recommended to be a powerful methodfor optimal power flow study on transmission power networks with considering the presence ofrenewable energies.

Keywords: coyote optimization algorithm; modified coyote optimization algorithm; transmissionpower system; success rate; photovoltaic systems

1. Introduction

Optimal power flow (OPF) is a very important optimization problem in power system operationdue to its huge contribution to economy and stable operation issues. The task in the problem is todetermine the most appropriate values for parameters of electric components in transmission powernetworks so that all operation limitations of the power system are always in the allowable ranges and

Energies 2019, 12, 4310; doi:10.3390/en12224310 www.mdpi.com/journal/energies

http://www.mdpi.com/journal/energies

http://www.mdpi.com

https://orcid.org/0000-0001-9312-0025

https://orcid.org/0000-0002-0951-410X

http://www.mdpi.com/1996-1073/12/22/4310?type=check_update&version=1

http://dx.doi.org/10.3390/en12224310

http://www.mdpi.com/journal/energies

Energies 2019, 12, 4310 2 of 36

predetermined objectives are also optimized [1,2]. Basically, single objective functions considered inthe OPF problem are fuel cost of generators, power losses of all branches, deviation of load voltage,and index of voltage stability while taking into account operation constraints of all componentssuch as active and reactive power output limitations of generators, tap limitation of transformers,reactive power output limitations of switchable capacitor banks, load bus voltage limitations, andtransmission line capacity limits. In the OPF problem, there are many parameters corresponding toworking parameters of electric components in transmission power networks. The parameters aredivided into two variable types, control variables and dependent variables. The former includes activepower of generators excluding the generator with the highest capacity located at the slack bus, voltageof generators, reactive power generation of shunt capacitors, and tap position of transformers. Thelatter includes other remaining parameters such as active power of the generator located at the slackbus, reactive power of generators, apparent power of all conductors, and voltage of all load buses.In summary, the key work in the OPF problem is to select control parameters of power networks inorder to satisfy all constraints and optimize predetermined objectives.

In the past, the mathematical model of the OPF problem did not consider non-differentiableobjectives as well as complicated constraints of generators. Therefore, the OPF problem was easilysolved by conventional deterministic optimization algorithms, which were mainly based on derivativesand gradient techniques. Nowadays, the OPF problem becomes a complex problem under consideringnon-convex or non-smooth objectives, and a high number of branches and generators. Therefore,the applications of the deterministic optimization algorithms must be stopped and replaced with otherpowerful algorithms. This issue boosts big motivations from researchers to propose new methods withthe main purpose of tackling such shortcomings. Metaheuristic algorithms comprising of standardmethods, improved methods, and hybrid methods were applied for coping with the OPF problem.Modified evolutionary programming (MEP) has been proposed for solving the OPF problem [3]. MEPwas the combination of conventional evolutionary programming (EP) and conventional simulatedannealing algorithm (SA). In the method, the mutation technique was established by using bothGaussian and Cauchy distributions whilst the selection technique was done by using the probabilisticupdating technique of SA. MEP was compared to EP in solving the IEEE 30-bus transmission powernetwork. Fuel cost comparison indicated that MEP was more effective than EP in terms of the optimalsolution. However, MEP had a longer execution time, although the two methods have been runby setting the same values for population size and the number of iterations. Differential evolutionvariants, such as conventional differential evolution (DE) [4,5], differential evolution with augmentedLagrange multiplier method (DE-ALM) [6], improved differential evolution (IDE) [7,8], were employedfor dealing with OPF problem. In [4], the search ability of DE has been tested on the OPF problem withtwo power systems considering single objective function and multi-objective function. In addition, theGrey Wolf algorithm has been implemented for the same study cases with the intent to demonstratethe high performance of DE for the OPF problem. DE has been evaluated to be more effective thanGWOA and other existing methods such as PSO, TLO, and GSA; however, the authors have forgotconvergence speed comparison, which is the most important factor for leading the conclusion of realperformance. As reported in [4], both DE and GWOA were run by setting 300 and 1000 to the highestnumber of iterations for IEEE 30-bus system and IEEE 118-bus system, but population size was notshown for the two methods. Clearly, the execution of DE was only for good results rather than thedemonstration of a high performance of DE. In [5], the authors have also tried to show the effectivenessof DE by running DE on an IEEE 30-bus system with different objective functions. The study hasset different values for the number of iterations although only the IEEE 30-bus system was the testcase. For the case with the good result, the number of iterations was set to 200, but for the case withworse results it was increased to 500 iterations. The settings in [5] obtained good results, but for thedemonstration of a real performance of DE. In [6], the authors have stated DE was not effective for theOPF problem because mutation factor was set to a fixed value and its ability for handling inequalityconstraint was not highly effective. Therefore, in DE-ALM, mutation factor and crossover factor have

Energies 2019, 12, 4310 3 of 36

been considered as control variables. Additionally, augmented Lagrange multiplier method (ALM) wasapplied for dealing with inequality constraints. DE-ALM has been compared to other metaheuristicalgorithms but the comparisons with DE have been ignored. Clearly, the proposed modificationsin [6] could not lead to any conclusions of the improved performance of DE-ALM. In contrast withDE-ALM, IDE methods [7,8] have only focused on improvements of mutation stage by employingdissimilar models for updated step size. In [7], IDE and conventional DE have been tested on IEEE6-bus and IEEE 30-bus systems for comparisons. In [8], larger systems with 30 buses and 57 buses havebeen used as study cases for proving the effectiveness of IDE. The two studies in [7,8] have focusedon the minimum fitness function but ignored the comparison of population size and the maximumiterations. Therefore, the conclusion of high performance of IDE was not persuasive. In addition toDE variants, particle swarm optimization (PSO) variants were also successfully applied for the OPFproblem considering different objectives and complicated constraints. These methods consist of PSOwith chaos queues and self-adaptive mechanisms (IPSO) [9], evolving ant direction particle swarmoptimization (EADPSO) [10], PSO with Pseudo-Gradient theory (PG-PSO) [11], graphics processingunit based PSO (GPUA-PSO) [12], and PSO with canonical differential evolution (CDE-PSO) [13]. In [9],the authors proved the outstanding performance of IPSO over conventional PSO through an IEEE30-bus system considering multi-objective function. These authors have found a set of non-dominatedsolutions and chosen one out of the solutions to be the best compromise solution. There was no reportof control parameters for search speed comparison. The shortcoming in [9] was tackled in [10] since thedemonstration of search speed was carried out. EADPSO and PSO have been run in the same settingof control parameters and in different settings for further comparisons. However, the study caseswere two systems with 39 buses and 57 buses. Furthermore, the comparisons between EADPSO andprevious methods were carried out based on objective functions only. Among these methods, PG-PSOin [11] was the most promising optimization tool thanks to the applications of the pseudo-gradientapproach and constriction factor. In PG-PSO, pseudo-gradient was in charge of finding the bestdirection for each particle in order to speed up its convergence process. In addition, the constrictionfactor could support finding a reasonable distance between the old solution and new solution. Via thetest results from benchmark functions and standard IEEE transmission power networks with 14, 30,57, and 118 buses, the proposed method has been stated to be more effective than the conventionalPSO. The studies in [12,13] have also tried to prove the real effectiveness of the applied methods byshowing less fuel cost and less total power loss than other methods; however, the setting of iterationshas not been considered for comparisons. In fact, the number of iterations was set to 3000 in [12] and1000 in [13] for an IEEE 30-bus system whereas the settings for other remaining systems were notreported. Aside from DE and PSO variants, genetic algorithm (GA) variants were thoroughly exploitedfor the OPF problem by testing the conventional genetic algorithm and analyzing its shortcomings.Different modified versions of GA have been employed such as genetic algorithm (GA) with fuzzysystem (FS-GA) [14], enhanced GA with new decoupled quadratic load flow (DQLF-EGA) [15], efficientGA with incremental power flow model (EGA-IPL) [16], and efficient GA with boundary method(EGA-BM) [17]. In [14], FS-GA has been compared to GA via three test systems with 6, 26, and 30 buses.The study has demonstrated the real effectiveness of the combination of GA and fuzzy mechanism, butit has ignored the comparison between FS-GA and other methods. In [15], DQLF-EGA and severalPSO methods have been implemented for only the IEEE 30-bus system whereas the comparisons withother methods have not been considered. However, both FS-GA and DQLF-EGA have been consideredas favorable methods for the OPF problem in [14,15]. In the GA method group, HEGA-BM was themost complicated method because it was created by combining HEGA [16] and a proposed boundarymethod. The strong point of such method was to solve all violations of upper and lower boundariesof OPF easily and successfully. HEGA-BM has provided better solutions than EGA-IPL for systemswith 30 and 118 buses. However, the main limitation in [16] was that the comparisons with otherexisting methods were few. In fact, just conventional methods like PSO, GA, and DE were compared toHEGA-BM. In addition to above popular methods, a huge number of methods consisting of original

Energies 2019, 12, 4310 4 of 36

methods, improved methods, and hybrid methods have also been suggested to cope with big challengesfrom the OPF problem [18–38]. IHBMO [20] was proposed in 2011 based on the honey bee matingoptimization algorithm (HBMO) by using the mutation mechanism. The method could effectivelyhandle some drawbacks of HBMO such as high probability of finding local optima and convergingto global optima with long simulation time. Unlikely, ISSO [38] was the latest tool and formed fromthe social spider optimization (SSO) by suggesting some improvements such as employing a newmovement method of female and male spiders and proposing a suitable ratio of female spiders tomale spiders. With the use of the proposed improvements, ISSO has found more impressive solutionsthan SSO and other methods. Among the studies, multi-objective function consisting of two singleobjectives such as fuel cost and power loss, power loss and voltage deviation, and fuel cost and voltagedeviation were considered in [18,19,21,23,24,27,30,33]. The main limitation in the studies was thetrade-off between two objectives since one objective was better but another was worse. Furthermore,the set of non-dominated solutions was not high enough to determine the best compromise solution.Almost all methods were run on an IEEE 30-bus system whereas few methods were run on an IEEE118-bus system. For leading to a conclusion of performance for the multi-objective problem, thesestudies have only focused on fuel cost, power loss, and voltage deviation but convergence speedreflected via the setting of control parameters. Clearly, the real performance of methods for the OPFproblem was not demonstrated persuasively and their key shortcomings were as follows:

1 They have compared a proposed method with other existing methods shown in other studiesmeanwhile they have forgot to demonstrate the improvement of the proposed method over thestandard method.

2 They have compared a proposed method with standard method, but they have forgot to comparethe method with other methods.

3 For the studies that compare the proposed method to both standard method and other existingmethods, the comparisons were only about objective functions not about convergence speed.

In the paper, we have tackled the disadvantages by comparing the proposed method (MCOA)with the standard method (OCOA) and other existing methods. In addition, the conclusion of realperformance of MCOA was established by taking into account the comparison of obtained results andthe setting of control parameters.

In recent years, many metaheuristic algorithms have been developed for the purpose ofsolving complicated optimization problems. Among these methods, original Coyote optimizationalgorithm (OCOA) in [39] is one of the most interesting methods with strange structure and veryhigh performance as compared to many metaheuristic algorithms. Forty functions with differentdimensions corresponding to 92 study cases have been employed to implement OCOA and six popularmetaheuristic algorithms such as PSO, ABCA, SOSA, BA, GWOA, and FA. OCOA could improve theresult over these six methods significantly once OCOA has found a better average fitness functionfor many test cases, especially for hybrid benchmark functions with high dimensions, which wererepresentatives of complicated optimization problems. Recently, OCOA has been successfully appliedfor different optimization problems in these studies [40–43]. In [40], OCOA has been applied forfinding main parameters of three-diode based photovoltaic modules under different environmentconditions consisting of the change of temperature and the change of irradiation. Thanks to the highperformance of OCOA, the three-diode photovoltaic model could be established and it could work aseffectively as other commercial Photovoltaic modules in the market. In [41], OCOA has been appliedfor reducing gas consumption of turbines in combined cycle power plants in Brazil while satisfyingpollutant emission regulations and physical limitations of turbines. OCOA has been evaluated to bemore effective than ABCA, BSA, SaDE, GWOA, PSO and the Symbiotic Organisms Search (SOS). In [42],OCOA together with GA and PSO have been implemented for solving the economic dispatch problemwith the integration of wind turbines and thermal power plants. The first system with five thermalpower plants and one wind turbine and the second system with 10 thermal power plants and two

Energies 2019, 12, 4310 5 of 36

wind turbines have been studied for comparisons. The target of the work was to minimize total cost ofthermal power plants and total cost of wind turbines. OCOA has reached less cost than GA and PSO forthe two systems. In [43], OCOA was applied for solving the reactive power dispatch problem in powersystem optimization operation; however, OCOA was less effective than other compared methodssince quality of voltage of load buses and total power losses in transmission lines were worse thanthose of other methods. Although OCOA has been applied successfully for different problems, its realperformance is still not demonstrated persuasively. Among the studies in [39–43], only optimal reactivepower dispatch problem in [43] is a real challenge for evaluating the effectiveness of OCOA. The resultcomparisons in [43] indicated that OCOA could not reach good results as other methods, even thesuccess rate of OCOA was very low. As observing results from the first application of OCOA onbenchmark function in [39], the most optimal solutions of the considered function set had “zero” valuewhereas OCOA has used a center solution in the first new solution generation technique. The centersolution approximately owns variables with middle point of upper bound and lower bound. Therefore,the center solution is very effective for OCOA in finding new solutions as solving benchmark functions.This regard is completely not effective for the OPF problem. In the second new solution generationtechnique, OCOA has used three random variables where two ones are taken from two randomlychosen existing solutions and another one is randomly produced within the upper and lower bounds.At the end of each iteration, OCOA performs solution exchange action among available groups. Theaction is useful for avoiding lumping solutions in each group; however, it cannot happen certainlydue to the comparison condition between a random number and a predetermined number. Clearly,randomizations are existing in OCOA. Due to the existing shortcomings of the two new solutiongeneration techniques and the solution exchange technique, OCOA cannot be a promising methodfor the OPF problem. It cannot find high quality solutions effectively and it must use a high numberof computation steps. For avoiding these disadvantages of OCOA, we propose modified coyoteoptimization algorithm (MCOA) by applying three modifications on the two new solution generationtechniques and the solution exchange technique. In the first modification, we suggest replacing thecenter solution with the best solution. In the second modification, we cancel randomizations of thesecond technique by searching nearby the best solutions. In the third modification, the action ofexchanging solutions among available groups is certainly performed by canceling the comparisoncondition. Consequently, the proposed modifications can bring the contributions to the proposedMCOA method as follows:

1 The replacement of the center solution is useful in reducing computation steps and shorteningsimulation time,

2 The replacement of randomizations in the second new solution technique can reduce the negativeimpact of randomizations and give more chances for finding more promising solutions,

3 Quitting condition of the solution exchange to make certainty of solution exchange amonggroups. This can avoid all solutions a group falling into local optimal zones and also reducecomputation steps.

In order to test the effectiveness and robustness of the proposed MCOA, three IEEE transmissionpower systems with 30 buses, 57 buses, and 118 buses are utilized as main study cases. The maintask of the methods is to determine control parameter of transmission power networks meanwhilethe objective is to minimize total fuel cost of all thermal generating units and total power losses in alltransmission lines. Furthermore, for the case of the IEEE 30-bus system, one photovoltaic system withrated power of 2.0 MW is proposed to be installed in order to reduce fuel cost of all thermal generatingunits and power loss in all transmission lines. As comparing obtained results, the main contributionsof the study are as follows:

1 MCOA can reach a higher success rate than OCOA as solving the OPF problem, especially for thelargest scale system with 118 buses,

2 MCOA can find better optimal solutions than OCOA for all study cases,

Energies 2019, 12, 4310 6 of 36

3 OCOA and MCOA methods are successfully applied for the OPF problem with the presence of aphotovoltaic system,

4 The best site of photovoltaic system (PVS) in the transmission power network found by MCOAcan result in much smaller fuel cost and power loss,

5 MCOA is superior to other compared methods.

In addition to the introduction section, the other parts of the paper are organized as follows:the mathematical formulation of OPF problem with the presence of objective functions and powersystem constraints is given in Section 2. The entire search procedure of OCOA is expressed and thenthe proposed MCOA is clarified in Section 3. All computation steps of using the proposed MCOA forthe OPF problem are described in detail in Section 4. Simulation results and discussion of three IEEEtransmission power networks are presented in Section 5. Conclusions are added in Section 6.

2. Mathematical Formulation of the OPF Problem

In this part, the considered OPF problem is expressed in terms of mathematical formulations withthe presence of single objective functions and constraints. Two single objectives are considered to bereduction of electric generation cost and reduction of power losses. All constraints in transmissiongrids must be satisfied such as operating limits of all electric components and power balance at all loadbuses. The objective functions and constraints are presented in detail as follows.

2.1. Single Objective Functions

2.1.1. Electric Generation Fuel Cost Reduction

In the OPF problem, power sources are thermal power plants using fossil fuels for producingelectric and supplying to loads. Therefore, the target of reducing electric generation fuel cost isequivalent to the target of optimizing the generation of all thermal power plants. Basically, thermalpower plants’ characteristics are represented as the relationship of power output and electric generationfuel cost. The operating efficiency of these thermal power plants can be evaluated by the total fuel costfor the generated power. The target of reducing fuel cost can be mathematically formulated by thefollowing Equation (1) [5]:

MinimizeNoG∑k=1

EGFCk =NoG∑k=1

a1k + a2kPk + a3kP2k +

∣∣∣∣a4k sin[a5k(Pmin

k − Pk)]∣∣∣∣ (1)

where a1k, a2k, a3k, a4k, a5k are coefficients in electric generation fuel cost function of the kth thermalpower plant; EGFCk is the electric generation fuel cost of the kth thermal power plant; NoG is thenumber of thermal power plants in the considered transmission grid; Pk is the generated power ofthe kth thermal power plant. In the OPF problem, control variables and dependent variables of eachsolution are as follows:

CVari ={Pk,i, Vk,i, Tt,i, SVCc,i

}T; i = 1, . . . , NoS, (2)

DVari ={P1,i, VLl,i, Qk,i, MVAbr,i

}T; i = 1, . . . , NoS, (3)

where CVari and DVari are the control and dependent variable sets of the ith solution.Corresponding to each variable in the two sets, the definitions are as follows:

1 k = 2, . . . , NoG for Pk,i,2 k = 1, . . . , NoG for Vk,i,3 t = 1, . . . , NoT for Tt,i,4 c = 1, . . . , NoSVC for SVCc,i,

Energies 2019, 12, 4310 7 of 36

5 l = 1, . . . , NoL for VLl,i,6 k = 1, . . . , NoG for Qk,i,7 br = 1, . . . , NoBR for MVAbr,i.

2.1.2. Total Active Power Losses (TAPL) Reduction

Total power generated by all generators is supplied to loads via transmission lines. With thepresence of resistance and reactance, transmission lines cause active and reactive power losses in whichactive power loss is considered as a major objective in the OPF problem. In fact, the transmission linesaccount for a high rate in all electric components in the power systems. Therefore, the active powerlosses in all transmission lines must be optimized as an objective. The objective can be expressed bythe mathematical model below [4]:

Minimize TAPL =NoG∑k=1

Pk −

NoL∑j=1

PL j. (4)

2.2. Constraints of the OPF Problem

The constraints of the OPF problem are similar to the constraints of control variables and dependentvariables together with the constraint of power balance at all buses. The constraint of variables isthe limits of electric components in the considered transmission grid such as generator of thermalpower plants, static VAR compensator, tap changer of transformers, loads, and transmission branches.Among the components, the generator of thermal power plants is the most complicated due to theconsideration of active power, reactive power, and voltage. With respect to mathematical formulation,the constraint of electric components can be expressed in terms of inequalities whereas the constraintof power balance at all load buses can be expressed in terms of equalities. All the constraints are shownas follows.

2.2.1. Constraints of all Electric Components

The generator of thermal power plants is constrained by upper and lower bounds of voltage,active, and reactive power as follows [11]:

Pmink ≤ Pk,i ≤ Pmax

k ; k = 1, . . . , NoG, (5)

Qmink ≤ Qk,i ≤ Qmax

k ; k = 1, . . . , NoG, (6)

Vmin≤ Vk,i ≤ Vmax; k = 1, . . . , NoG. (7)

In addition, transformers, static VAR compensators, load, and transmission branches areconstrained by upper and lower bounds as follows [10]:

Tmin≤ Tt,i ≤ Tmax; t = 1, . . . , NoT, (8)

SVCminc ≤ SVCc,i ≤ SVCmax

c ; c = 1, . . . , NoSVC, (9)

VLmin≤ VLl,i ≤ VLmax; l = 1, . . . , NoL, (10)

MVAbr,i ≤MVAmaxbr ; br = 1, . . . , NoBR. (11)

Energies 2019, 12, 4310 8 of 36

2.2.2. Constraints of Active and Reactive Power Balance

Two equality constraints are considered in the considered OPF problem to be the active powerbalance and the reactive power balance at each bus in the transmission grid. The two constraints areshown as follows [23]:

P j = PL j + V j

NoB∑i=1

Vi[G ji cos( f j − fi) + B ji sin( f j − fi)

]; j = 1, . . . , NoB, (12)

Q j + SVC j = QL j + V j

NoB∑i=1

Vi[G ji sin( f j − fi) − B ji cos( f j − fi)

]; j = 1, . . . , NoB. (13)

As seen from the two equalities above, generated power Pj must be equal to the sum of the loaddemand PLj and the power supplying to the branch connecting bus j and bus i.

In this paper, we consider three benchmark power systems with 30, 57, and 118 buses that havethe same characteristics of a real system. In fact, the systems are comprised of transformers, capacitors,conductors, generators, loads, and buses. These components are represented by important parts andincluded in the OPF problem. For instance, transformers are represented as the tap changer andlimitations of the tap changer are considered in Equation (8). Capacitors are also concerned and installedat some buses with the intent to supplement reactive power to loads and control voltage of load buses.Limitations of reactive power from the capacitors are shown in Equation (9). Important parametersof loads are active power, reactive power, and working voltage limitations. Voltage limitations ofloads are shown in Equation (10) meanwhile the balances of active and reactive power are shownin Equations (12) and (13). Conductors or transmission lines are taken into account by the presenceof apparent power limits shown in Equation (11). In dealing with optimization problems in powersystems, the most difficult task is to collect parameters of electric components in real systems. Therefore,the solution is to use these benchmark systems that have the same characteristics of real systems.

3. The Proposed Method

3.1. Original Coyote Optimization Algorithm

OCOA was inspired from the natural behaviors of coyotes. The strange characteristics of OCOAare two new solution generation techniques and one more technique for exchanging solutions amonggroups. OCOA is totally different from other metaheuristic algorithms since the whole population isdivided into a number of groups (NoGr) and there is a number of members in each groups (NoCoy).Each coyote is represented by its social condition and quality of the social condition. Between the twofactors, the former is corresponding to the optimal solution and the latter is corresponding to the fitnessfunction value of the optimal solution. OCOA executes two techniques for producing two new solutiontimes. OCOA produces (NoGr × NoCoy) solutions in the first time but OCOA produces NoGr solutionsin the second time. It is noted that the result of (NoGr × NoCoy) is also population size. All solutions ineach group are newly updated in the first technique while only one solution in each group is newlyupdated in the second technique. At the last step of OCOA, two randomly picked solutions in tworandomly chosen groups can be exchanged based on a predetermined condition. Therefore, it seems tobe that OCOA is a promising method for the considered OPF problem. For better understanding ofOCOA, the whole computation steps of OCOA method are described in detail as follows [39]:

Similar to other existing meta-heuristic algorithms, the first stage in the search process of OCOA isinitialization with the target of producing the initial solutions. Each coyote in each group is a solutionof the optimization problem and it is represented as Coym,g where m is the mth coyote in each groupand g is the gth group. Each solution is randomly produced based on the following model:

Coym,g = LB + λ1(UB− LB); m = 1, . . . , NoCoy; g = 1, . . . , NoGr, (14)

Energies 2019, 12, 4310 9 of 36

where LB and UB are respectively lower and upper bounds of solutions. In addition, each solutioncontains NoCV control variables.

Each solution Coym,g is evaluated by calculating fitness function FTm,g and the best solution ofeach group is determined and set to Coybest,g.

3.2. Two New Solution Generations

Unlike PSO, which has one new solution generation, OCOA is provided with two techniquesof generating new solutions for each iteration. Each group produces NoCoy solutions for the firstgeneration but produces only one new solution for the second generation. The first technique isperformed by the meaning of the following equation:

Coym,g = Coym,g + λ2(Coybest,g −Coypr1,g) + λ3(Coycen,g −Coypr2,g). (15)

In Equation (15), Coym,g on the left hand side is the newly updated solution meanwhile Coym,g

on the right hand side is the old solution. In case that Coym,g on the left hand side and Coym,g on theright hand side are the same, it means the solution is not newly updated. However, the possibility ofthe situation is very low or nearly equal to zero. In order to determine the center solution Coycen,g,one by one variable in the gth group is sorted in descending order of values and then the middle valueis selected for the considered variable.

For the second generation, OCOA selects one out of three ways for taking control variables for thesole solution. Each group produces one solution Coyg, which is represented as follows:

Coyg =[varn,g

]; n = 1, . . . , NoCV, (16)

where varn,g is determined by the following rule:

varn,g =

varn,pr1,g if λ4 <

1NoCV

varn,pr2,g if λ4 <1

NoCV + 0.5varn,rd if ortherwire

, (17)

where varn,pr1,g and varn,pr2,g are two variables that are taken from the first randomly picked solutionand the second randomly picked solution, respectively. If the first condition, λ4 <

1NoCV , is true,

varn,pr1,g is selected but if the second condition, λ4 <1

NoCV + 0.5, is true, varn,pr2,g is selected. In thecase that both of the conditions are incorrect, varn,rd is randomly produced within the lower andupper bounds.

3.3. Correction of New Solutions

New solutions Coym,g and Coyg in Equations (15) and (16) need verification and correction beforeevaluating quality. The verification and correction can be carried out by the function of the followingmodels:

Coym,g =

UB if Coym,g > UBLB if Coym,g < LB; m = 1, . . . , NoCoy; g = 1, . . . , NoGrCoym,g else

, (18)

Coyg =

UB if Coyg > UBLB if Coyg < LB; g = 1, . . . , NoGrCoyg else

. (19)

Energies 2019, 12, 4310 10 of 36

3.4. Solution Exchange among Available Groups

OCOA terminates each iteration by performing solution exchange among available groups.However, the action is not always performed and dependent on the condition below:

λ5 < 0.005×NoCoy2. (20)

If the condition in Equation (20) occurs, two randomly picked solutions in two random groupsare exchanged together. Otherwise, all solutions are retained in their current position.

3.5. Modified Coyote Optimization Algorithm (MCOA)

3.5.1. The First Proposed Modification

In the first proposed modification, we suggest changing Equation (15) by replacing Coycen,g withCoyGbest and replacing Coypr1,g and Coypr2,g with Coym,g. As a result, all solutions in the gth group areproduced by the following formula:

Coym,g = Coym,g + λ1(Coybest,g −Coym,g) + λ2(CoyGbest −Coym,g). (21)

In Equation (21), the center solution was removed due to its negative impacts on the qualityof new solutions of the OPF problem. The OCOA in [39] was applied for benchmark optimizationfunctions, meanwhile approximately all functions had an optimal solution with zero values. Thus,the center solution was highly effective for generating updated steps and new solutions for OCOAwith benchmark functions. In the paper, optimal solutions of the OPF problem has a set of controlvariables with different values. For instance, power output of generators is in the range of higher than0 and can be up to 1000 MW, meanwhile reactive power of static VAR compensators can be up totens of MVAR. Unlikely, the tap changer of transformers and load bus voltage have values around1.0 Pu. Consequently, center solutions can negatively influence the performance of OCOA for optimalsolutions of the OPF problem. In addition, Equation (21) has employed the old solution Coym,g insteadof two randomly picked solutions Coypr1,g and Coypr2,g like Equation (15). The modification can assurethat all solutions in the considered group are used to generate updated steps and new solutions.

3.5.2. The Second Proposed Modification

In the second improvement, Equation (17) is suggested to be modified due to its low effectivenessof randomization in producing control variables of the sole new solution. The first control variable isselected by the comparison between a random number and (1/NoCV) while the second control variableis determined based on the comparison between the random number and (1/NoCV + 0.5). If the firsttwo conditions are not met, a random control variable is produced within upper and lower bounds.Clearly, the combination of the three randomizations cannot lead to a good solution with high qualityexcluding the diversification of generated control variables. For avoiding missing a promising solution,we suggest Equations (16) and (17) should be canceled and replaced with a new one that can supportgenerating a more favorable solution. The new formula is as follows:

Coyg = Coybest,g + λ6(CoyGbest −Coybest,g) + λ7(Coypr1,g −Coybest,g). (22)

3.5.3. The Third Proposed Modification

In the third improvement, we suggest ignoring the comparison condition for exchanging solutionsshown in Equation (20). Regarding the review on the action of exchanging solutions among groups,the action can diversify the solutions in each group. Thus, we select two random groups and exchangetwo randomly picked solutions in the groups.

Energies 2019, 12, 4310 11 of 36

4. Implementation of the Proposed Method for the OPF Problem

4.1. Population Initialization

OPF is a nonlinear problem with a high number of control variables and it needs to solve theproblem simply and easily. In the study, we used the proposed MCOA method to find good controlparameters of transmission power networks and then the found control parameters were added intothe Matpower program where Newton-Graphson was run to find remaining dependent parameters oftransmission power networks consisting of generation of slack bus, voltage of all load buses, reactivepower of generators, and branch currents. Each solution of MCOA contains a set of control variablesthat are parameters in transmission power network such as voltage and active power of generators,working reactive power of compensators, and secondary voltage of transformers. For evaluating thequality of each solution, the sum of the considered objective function and the violation of all dependentparameters is calculated. In the first step, the set of control variables consisting of Pk,i (k = 2, . . . ,NoG), Vk,i (k = 1, . . . , NoG), Tt,i (t = 1, . . . , NoT) and SVCc,i (c = 1, . . . , NoSVC) is randomly producedas follows:

Pk,i = Pmink + λ8(Pmax

k − Pmink ); k = 2, . . . , NoG; i = 1, . . . , NoS, (23)

Vk,i = Vmin + λ9(Vmax−Vmin); k = 1, . . . , NoG; i = 1, . . . , NoS, (24)

Tt,i = Tmin + λ10(Tmax− Tmin); t = 1, . . . , NoT; i = 1, . . . , NoS, (25)

SVCc,i = SVCminc + λ11(SVCmax

c − SVCminc ); c = 1, . . . , NoSVC; i = 1, . . . , NoS. (26)

4.2. Solution Quality Evaluation

After determining all control variables as shown in Equations (23)–(26), such control variables aresent to input data of the Matpower program for execution and obtaining other dependent variablessuch as P1,i, VLl,i, Qk,i, and MVAbr,i. As a result, quality evaluation of the ith solution is completed bycalculating the fitness function. In the paper, two independent optimization cases are considered to bereduction of electricity generation fuel cost and the reduction of total active power losses. Accordingto the two cases, the fitness function is mathematically modeled as follows:

FTi =NoG∑k=1

EGFCk,i + C1(∆P1,i) + C2

NoL∑l=1

∆VLl,i

+ C3

NoG∑k=1

∆Qk,i

+ C4

NoBR∑br=1

∆Qbr,i

, (27)

FTi = TAPLi + C1(∆P1,i) + C2

NoL∑l=1

∆VLl,i

+ C3

NoG∑k=1

∆Qk,i

+ C4

NoBR∑br=1

MVAbr,i

, (28)

where C1, C2, C3, and C4 are penalty factors and selected by experiment. The factors are set to 103, 104,and 105 for IEEE 30, 57, and 118-bus systems, respectively; and ∆P1,i, ∆VLl,i, ∆Qk,i, and ∆MVAbr,i arepenalty terms corresponding to violations of dependent variables. The penalty terms are obtained byusing the formulas below:

∆P1,i =

0 if Pmin

1 ≤ P1,i ≤ Pmax1

(Pmin1 − P1,i)

2 if Pmin1 > P1,i

(P1,i − Pmax1 )2 else

, (29)

Energies 2019, 12, 4310 12 of 36

∆VLl,i =

0 if VLmin

≤ VLl,i ≤ VLmax

(VLmin−VLl,i)

2 if VLmin > VL1,i

(VLl,i −VLmax)2 else, , (30)

∆Qk,i =

0 if Qmin

k ≤ Qk,i ≤ Qmaxk

(Qmink −Qk,i)

2 if Qmink > Qk,i

(Qk,i −Qmaxk )2 else

, (31)

∆MVAbr,i =

0 if MVAbr,i ≤MVAmaxbr

(MVAbr,i −MVAmaxbr )2 else

. (32)

4.3. Produce and Fix New Solutions

As shown in Equations (21) and (22), MCOA experiences two generations of new solutions.Therefore, there have to be two times for checking the violation of new solutions and correcting newsolutions. Each new solution is comprised of Pk,i (k = 2, . . . , NoG), Vk,i (k = 1, . . . , NoG), Tt,i (t = 1, . . . ,NoT), and SVCc,i (c = 1, . . . , NoSVC). Therefore, these variables must be checked and corrected byusing the following formulas:

Pk,i =

Pk,i if Pk,i ∈

[Pmin

k , Pmaxk

]Pmin

k if Pk,i ∈[− ∞, Pmin

k

]Pmax

k if Pk,i ∈[Pmax

k ,+∞] ; k = 2, . . . , NoG; i = 1, . . . , NoS, (33)

Vk,i =

Vk,i if Vk,i ∈

[Vmin, Vmax

]Vmin if Vk,i ∈

[− ∞, Vmin

]Vmax if Vk,i ∈ [Vmax,+∞]

; k = 2, . . . , NoG; i = 1, . . . , NoS, (34)

Tt,i =

Tt,i if Tt,i ∈

[Tmin, Tmax

]Tmin if Tt,i ∈

[− ∞, Tmin

]Tmax if Tt,i ∈ [Tmax,+∞]

; t = 1, . . . , NoT; i = 1, . . . , NoS, (35)

SVCc,i =

SVCc,i if SVCc,i ∈

[SVCmin

c , SVCmaxc

]SVCmin

c if SVCc,i ∈[− ∞, SVCmin

c

]SVCmax

c if SVCc,i ∈ [SVCmaxc ,+∞]

; c = 1, . . . , NoSVC; i = 1, . . . , NoS. (36)

4.4. The Whole Procedure of Solving the OPF Problem

MCOA is a population based metaheuristic algorithm. Therefore, MCOA stops searching newsolutions if the current iteration can reach the maximum value. The whole search process of using theproposed MCOA method for solving the OPF problem can be described as the following steps.

• Step 1: Select control variables and set values to the variables for the current population.• Step 2: Run Matpower program to find dependent variables.• Step 3: Calculate fitness function to determine the global best solution for all group and the local

best solution for each group.• Step 4: Produce new solutions for all groups and fix new solutions if the solutions violate lower or

upper bounds.• Step 5: Run Matpower program to find dependent variables.• Step 6: Calculate fitness function of new solutions. Compare new and old solutions to retain

better ones.

Energies 2019, 12, 4310 13 of 36

• Step 7: Produce one new solution for each solution and fix it if it violates lower or upper bounds.• Step 8: Choose the worst solution for each group.• Step 9: Run Matpower program to find dependent variables.• Step 10: Calculate fitness function of the new solution. Compare the new solution and the worst

solution in each group to retain the better one.• Step 11: Exchange solutions among the available groups.• Step 12: Determine the best solution of the whole population.

The process with 12 steps above is included in one iteration in the iterative algorithm of solvingthe OPF problem by using the MCOA method, meanwhile the detail of application for the wholesearch of one trial run can be implemented as shown in Figure 1.Energies 2019, 12, x FOR PEER REVIEW 14 of 36

Iter NoIter= 1Iter Iter= +

Figure 1. The iterative algorithm of solving the optimal power flow (OPF) problem by using the modified coyote optimization algorithm (MCOA) method.

5. Simulation Results and Discussions

This section presents the simulation results from solving three transmission power systems with 30, 57, and 118 buses by using OCOA and the proposed MCOA. The whole data of the systems can be reached by referring to [11], meanwhile the coefficients of electric generation fuel cost function of IEEE 30-bus system are taken from [5]. In addition, the whole data are also supplemented together with the paper. For each considered objective, 50 successful runs were performed for obtaining 50 fitness functions corresponding to 50 optimal solutions. Then, main factors consisting of minimum, mean, and maximum of fitness function are reported for comparison. The implementation of the OCOA and the proposed MCOA was executed on the Matlab program language with version R2016a and personal computer with CPU: Intel Core i7-2.4GHz-RAM 4GB, GPU: Intel HD Graphics 5500, and operating system version: Windows 8.1 Pro-64-bit.

5.1. Result Comparisons between OCOA and MCOA for IEEE 30-Bus Transmission Power Network

Figure 1. The iterative algorithm of solving the optimal power flow (OPF) problem by using themodified coyote optimization algorithm (MCOA) method.

Energies 2019, 12, 4310 14 of 36

5. Simulation Results and Discussions

This section presents the simulation results from solving three transmission power systems with30, 57, and 118 buses by using OCOA and the proposed MCOA. The whole data of the systems can bereached by referring to [11], meanwhile the coefficients of electric generation fuel cost function of IEEE30-bus system are taken from [5]. In addition, the whole data are also supplemented together withthe paper. For each considered objective, 50 successful runs were performed for obtaining 50 fitnessfunctions corresponding to 50 optimal solutions. Then, main factors consisting of minimum, mean,and maximum of fitness function are reported for comparison. The implementation of the OCOA andthe proposed MCOA was executed on the Matlab program language with version R2016a and personalcomputer with CPU: Intel Core i7-2.4GHz-RAM 4GB, GPU: Intel HD Graphics 5500, and operatingsystem version: Windows 8.1 Pro-64-bit.

5.1. Result Comparisons between OCOA and MCOA for IEEE 30-Bus Transmission Power Network

5.1.1. Comparisons between OCOA and MCOA Methods for the Case without PVS

In this section, we compare the real performance of the proposed MCOA method with OCOA andother existing methods in finding optimal solutions for the IEEE 30-bus transmission power networkwith the two optimization cases consisting of power losses and fuel cost. For pair comparisons betweenthe proposed method and OCOA, we set control parameters to the same values. However, the faircomparisons between the proposed method and other ones are based on the number of solution qualityevaluations as well as the best fitness function values. Basically, the number of fitness evaluationsfor each trial run is also equivalent to the number of times that Matpower program had to be run.In the OPF problem, all dependent variables of each solution must be found and then both dependentvariables and control variables are used to calculate the fitness function.

For the best observation of the performance comparisons between OCOA and MCOA, we executedOCOA and MCOA by setting different values to NoGr, NoCoy but fixing NoIter = 100. Tables 1 and 2report the results with three different settings for the cases of minimizing total fuel cost and total activepower losses. Fitness function values of 50 successful runs are summarized by using minimum fitness,mean fitness, maximum fitness, and standard deviation. In addition to the results, the improvement ofminimum fitness (IoMF) from the proposed method over OCOA and success rate (SR) in percent fromthe two methods are also calculated by using the following equations:

IoMF(%) =Min. fitness of OCOA − Min. fitness of MCOA

Min. fitness of OCOA× 100%, (37)

SR(%) =Number of successful runs

Total number of runs× 100%. (38)

Thus, improvement of minimum fitness and SR are also seen in the tables for a better view ofperformance comparison. The values of minimum fitness indicate that MCOA can find a much betteroptimal solution than OCOA for the different settings. For the cost objective, the best fitness of OCOAand MCOA are, respectively, $842.871 and $818.330 for the setting of NoGr = 3, NoCoy = 3, NoIter = 100,$829.244 and $799.899 for the setting of NoGr = 3, NoCoy = 4, NoIter = 100, and $801.643 and $798.916for the setting of NoGr = 4, NoCoy = 4, NoIter = 100. Clearly, the proposed MCOA method can reach abetter solution than OCOA for all the setting cases. The outstanding solutions can be evaluated byobserving the improvement of minimum fitness that are 2.91%, 3.54%, and 0.34% corresponding to thethree setting cases. Similarly, the proposed MCOA method also gets better mean fitness, maximumfitness, and standard deviation than OCOA for the three setting cases. Table 2 also indicates highperformance of the proposed method over OCOA once the proposed method can reach much smallervalues of total active power losses and the improvement of the minimum fitness is up to 29.45%,30.21%, and 28.05% for three setting cases. Other values from the proposed method such as meanfitness, maximum fitness, and standard deviation are also less than those from OCOA. In the two

Energies 2019, 12, 4310 15 of 36

tables, SR from MCOA is always higher than that from OCOA. In fact, MCOA can reach SR with100% for the settings of NoGr = 3, NoCoy = 4, NoIter = 100 and NoGr = 4, NoCoy = 4, NoIter = 100while the best SR from OCOA is 88% for the setting of NoGr = 4, NoCoy = 4, NoIter = 100. Therefore,MCOA outperforms OCOA in terms of constraint handling ability.

Table 1. Results obtained by the original coyote optimization algorithm (OCOA) and MCOAfor the EGFC objective of the IEEE 30-bus transmission power network with different settings ofcontrol parameters.

Fitness Values over50 Successful Runs

NoGr = 3, NoCoy = 3,NoIter = 100



OCOA MCOA OCOA MCOA OCOA MCOA

Min. fitness 842.871 818.330 829.244 799.899 801.643 798.916Mean fitness 846.024 819.628 840.338 810.026 812.368 800.184Max. fitness 929.852 822.835 886.162 812.391 856.667 803.314

Std. deviation 10.493 1.112 8.785 1.099 8.493 1.086SR (%) 68 86 82 100 86 100

IoMF (%) 2.91 3.54 0.34

Table 2. Result obtained by OCOA and MCOA for the total active power losses (TAPL) objective of theIEEE 30-bus transmission power network with different settings of control parameters.

Fitness Values over50 Successful Runs




OCOA MCOA OCOA MCOA OCOA MCOA

Min. fitness 4.125 2.910 4.085 2.851 3.957 2.847Mean fitness 5.480 3.272 5.454 3.233 5.277 3.194Max. fitness 15.338 5.061 14.727 4.997 14.277 4.941

Std. dev fitness 1.642 0.423 1.584 0.422 1.528 0.414SR (%) 69 88 80 100 88 100

IoMF (%) 29.45 30.21 28.05

For a better view of outstanding performance of the proposed method over OCOA, 50 fitnessfunction values of 50 successful runs and the convergence characteristic of the best run are plotted inFigures 2 and 3 for the cost objective and in Figures 4 and 5 for the loss objective. The figures are resultsfrom the setting of NoGr = 4, NoCoy = 4 and NoIter = 100. Figures 2 and 4 show that the red curve hasvery tiny fluctuations and many red points have the same fitness as the best red point, while the bluecurve has very high fluctuations and few blue points have smaller fitness than the worst red point.Figures 3 and 5 point out the significantly faster search speed of the proposed method whereas OCOAis very slow. It can be seen that the fitness of the proposed method at the 20th iteration is much smallerthan that of OCOA at the final iteration. Therefore, it can be stated that the proposed MCOA method issuperior to OCOA in terms of:

1 Finding much better optimal solutions.2 Reaching more stable search ability.3 Converging to high quality solutions with significantly faster manner.

Energies 2019, 12, 4310 16 of 36

Energies 2019, 12, x FOR PEER REVIEW 16 of 36

Std. deviation 10.493 1.112 8.785 1.099 8.493 1.086 SR (%) 68 86 82 100 86 100

IoMF (%) 2.91 3.54 0.34

Table 2. Result obtained by OCOA and MCOA for the total active power losses (TAPL) objective of the IEEE 30-bus transmission power network with different settings of control parameters.

Fitness Values over 50 Successful Runs

NoGr = 3, NoCoy = 3, NoIter = 100



OCOA MCOA OCOA MCOA OCOA MCOA Min. fitness 4.125 2.910 4.085 2.851 3.957 2.847 Mean fitness 5.480 3.272 5.454 3.233 5.277 3.194 Max. fitness 15.338 5.061 14.727 4.997 14.277 4.941

Std. dev fitness 1.642 0.423 1.584 0.422 1.528 0.414 SR (%) 69 88 80 100 88 100

IoMF (%) 29.45 30.21 28.05

For a better view of outstanding performance of the proposed method over OCOA, 50 fitness function values of 50 successful runs and the convergence characteristic of the best run are plotted in Figures 2 and 3 for the cost objective and in Figures 4 and 5 for the loss objective. The figures are results from the setting of NoGr = 4, NoCoy = 4 and NoIter = 100. Figures 2 and 4 show that the red curve has very tiny fluctuations and many red points have the same fitness as the best red point, while the blue curve has very high fluctuations and few blue points have smaller fitness than the worst red point. Figures 3 and 5 point out the significantly faster search speed of the proposed method whereas OCOA is very slow. It can be seen that the fitness of the proposed method at the 20th iteration is much smaller than that of OCOA at the final iteration. Therefore, it can be stated that the proposed MCOA method is superior to OCOA in terms of:

1 Finding much better optimal solutions. 2 Reaching more stable search ability. 3 Converging to high quality solutions with significantly faster manner.

Figure 2. The fitness function values of 50 successful runs obtained by OCOA and MCOA for the EGFC objective of the IEEE 30-bus transmission power network.

Fitn

ess

func

tion

Figure 2. The fitness function values of 50 successful runs obtained by OCOA and MCOA for the EGFCobjective of the IEEE 30-bus transmission power network.


Figure 3. Fitness function iteration characteristic of the best run from OCOA and MCOA for the EGFC objective of the IEEE 30-bus transmission power network.

Figure 4. The fitness function values of 50 successful runs obtained by OCOA and MCOA for the TAPL objective of the IEEE 30-bus transmission power network.

Figure 5. Fitness function iteration characteristic of the best run from OCOA and MCOA for TAPL objective of the IEEE 30-bus transmission power network.

Figure 3. Fitness function iteration characteristic of the best run from OCOA and MCOA for the EGFCobjective of the IEEE 30-bus transmission power network.





Figure 4. The fitness function values of 50 successful runs obtained by OCOA and MCOA for the TAPLobjective of the IEEE 30-bus transmission power network.

Energies 2019, 12, 4310 17 of 36





Figure 5. Fitness function iteration characteristic of the best run from OCOA and MCOA for TAPLobjective of the IEEE 30-bus transmission power network.

5.1.2. Comparisons between OCOA and MCOA Methods for the Case with PVS

In this section, we survey the real performance of OCOA and MCOA on a more complicatedcase of IEEE 30-bus system. A photovoltaic system with rated power of 2.0 MW was decided to beinstalled in the system. We suppose that geographical location of buses 3, 7, 8, 15, 18, 19, 20, 21, 22, 24,26, 29, and 30 is appropriate for installing the PVS with power of 2.0 MW. We suppose the PVS uses aninverter to convert DC into AC without battery storage systems. Therefore, environment issue is not aproblem in the installation of PV system. Thus, the duty of OCOA and MCOA in this studied case is todetermine the location of installed PVS and other control variables of the conventional OPF problem.As a result, the control variable set for this studied case consists of SPi, Pk,i, Vk,i, Tt,i, and SVCc,i.

For implementing OCOA and MCOA for minimizing total power losses, we used the best settingof Section 5.1.1, i.e., NoGr = 4, NoCoy = 4, and NoIter = 100. The obtained results from the two methodsover 50 successful runs including minimum, mean, and maximum power losses are compared. Resultsof total fuel cost are plotted in Figures 6–8 meanwhile those of total power losses are plotted inFigures 9–11. Figures 6 and 7 indicate that OCOA and MCOA can reduce total fuel cost due to installingPVS. Namely, OCOA and MCOA can improve fuel cost by 0.7% and 0.88%. Similarly, Figures 9 and 10also show less power losses from OCOA and MCOA with PVS. Namely, OCOA and MCOA canimprove power loss by 8.5% and 3.8%. As comparing results from OCOA and MCOA for the case ofinstalling PVS, Figures 8 and 11 show better cost and better loss of MCOA, respectively. As comparedto OCOA, MCOA can improve fuel cost up to 0.51% and power loss up to 24.36%. Clearly, installingPVS can support OCOA and MCOA find better solutions with less total cost and less power losses.

The best location of PVS found by OCOA and MCOA is, respectively, at buses 30 and 15 for fuelcost objective, and at buses 22 and 30 for power loss objective. Figures 12 and 13 show the IEEE 30-bussystem with the installation of PVS found by MCOA.

Energies 2019, 12, 4310 18 of 36


5.1.2. Comparisons between OCOA and MCOA Methods for the Case with PVS

In this section, we survey the real performance of OCOA and MCOA on a more complicated case of IEEE 30-bus system. A photovoltaic system with rated power of 2.0 MW was decided to be installed in the system. We suppose that geographical location of buses 3, 7, 8, 15, 18, 19, 20, 21, 22, 24, 26, 29, and 30 is appropriate for installing the PVS with power of 2.0 MW. We suppose the PVS uses an inverter to convert DC into AC without battery storage systems. Therefore, environment issue is not a problem in the installation of PV system. Thus, the duty of OCOA and MCOA in this studied case is to determine the location of installed PVS and other control variables of the conventional OPF problem. As a result, the control variable set for this studied case consists of SPi, Pk,i, Vk,i, Tt,i, and SVCc,i.

For implementing OCOA and MCOA for minimizing total power losses, we used the best setting of Section 5.1.1, i.e., NoGr = 4, NoCoy = 4, and NoIter = 100. The obtained results from the two methods over 50 successful runs including minimum, mean, and maximum power losses are compared. Results of total fuel cost are plotted in Figures 6–8 meanwhile those of total power losses are plotted in Figures 9–11. Figures 6 and 7 indicate that OCOA and MCOA can reduce total fuel cost due to installing PVS. Namely, OCOA and MCOA can improve fuel cost by 0.7% and 0.88%. Similarly, Figures 9 and 10 also show less power losses from OCOA and MCOA with PVS. Namely, OCOA and MCOA can improve power loss by 8.5% and 3.8%. As comparing results from OCOA and MCOA for the case of installing PVS, Figures 8 and 11 show better cost and better loss of MCOA, respectively. As compared to OCOA, MCOA can improve fuel cost up to 0.51% and power loss up to 24.36%. Clearly, installing PVS can support OCOA and MCOA find better solutions with less total cost and less power losses.

The best location of PVS found by OCOA and MCOA is, respectively, at buses 30 and 15 for fuel cost objective, and at buses 22 and 30 for power loss objective. Figures 12 and 13 show the IEEE 30-bus system with the installation of PVS found by MCOA.

Figure 6. Comparison of total fuel cost obtained by OCOA for the cases with and without PVS.

801.643

812.368

856.667

795.982806.209

853.844

760770780790800810820830840850860870

Minimum fuel cost Mean fuel cost Maximum fuel cost

Tota

l fue

l cos

t ($/

h)

OCOA without PVS OCOA with PVS

Figure 6. Comparison of total fuel cost obtained by OCOA for the cases with and without PVS.Energies 2019, 12, x FOR PEER REVIEW 19 of 36

Figure 7. Comparison of total fuel cost obtained by MCOA for the cases with and without PVS.

Figure 8. Comparison of total fuel cost obtained by OCOA and MCOA for the case with PVS.

798.916 800.184 803.314

791.917796.481

855.737

760

780

800

820

840

860

880


Tota

l fue

l cos

t ($/

h)

MCOA without PVS MCOA with PVS

795.982806.209

853.844

791.917796.481

855.737

760

780

800

820

840

860

880


Tota

l fue

l cos

t ($/

h)

OCOA with PVS MCOA with PVS

3.9575.277

14.277

3.6214.857

7.252

0

2

4

6

8

10

12

14

16

Minimum power loss Mean power loss Maximum power loss

Tolta

l pow

er lo

sses

(MW

)



Energies 2019, 12, 4310 19 of 36




798.916 800.184 803.314

791.917796.481

855.737

760

780

800

820

840

860

880


Tota

l fue

l cos

t ($/

h)


795.982806.209

853.844

791.917796.481

855.737

760

780

800

820

840

860

880


Tota

l fue

l cos

t ($/

h)


3.9575.277

14.277

3.6214.857

7.252

0

2

4

6

8

10

12

14

16


Tolta

l pow

er lo

sses

(MW

)






798.916 800.184 803.314

791.917796.481

855.737

760

780

800

820

840

860

880


Tota

l fue

l cos

t ($/

h)


795.982806.209

853.844

791.917796.481

855.737

760

780

800

820

840

860

880


Tota

l fue

l cos

t ($/

h)


3.9575.277

14.277

3.6214.857

7.252

0

2

4

6

8

10

12

14

16


Tolta

l pow

er lo

sses

(MW

)


Figure 9. Comparison of total power losses obtained by OCOA for the cases with and without PVS.

Energies 2019, 12, 4310 20 of 36



Figure 10. Comparison of total power losses obtained by MCOA for the cases with and without PVS.

Figure 11. Comparison of total power losses obtained by OCOA and MCOA for the case with PVS.

2.8473.194

4.941

2.7393.147

5.06

0

1

2

3

4

5

6


Tota

l pow

er lo

sses

(MW

)


3.621

4.857

7.252

2.7393.147

5.06

0

1

2

3

4

5

6

7

8


Tota

l pow

er lo

sses

(MW

)







2.8473.194

4.941

2.7393.147

5.06

0

1

2

3

4

5

6


Tota

l pow

er lo

sses

(MW

)


3.621

4.857

7.252

2.7393.147

5.06

0

1

2

3

4

5

6

7

8


Tota

l pow

er lo

sses

(MW

)



Energies 2019, 12, 4310 21 of 36Energies 2019, 12, x FOR PEER REVIEW 21 of 36

GG

G G

GG

Figure 12. The best location of PVS found by MCOA for the case of reducing total fuel cost. Figure 12. The best location of PVS found by MCOA for the case of reducing total fuel cost.

Energies 2019, 12, 4310 22 of 36


GG

G G

GG

Figure 13. The best location of PVS found by MCOA for the case of reducing total power losses.

5.2. Comparisons of Result from MCOA and Other Method for the IEEE 30-Bus Power System

Comparisons of total fuel cost and total power losses from MCOA and other methods are shown in Tables 3 and 4, respectively. In addition to the best, mean, and worst values of cost and loss, three other factors including population size, the maximum number of iterations, and the number of solution quality evaluations are also mentioned for performance comparisons. Among the factors, population size and the number of solution quality evaluations of the proposed method are calculated by using the following equations:

Npop NoGr NoCoy= × , (39)

= × × + × ( ) ( 1)NoSqe NoIter NoGr NoCoy NpGr . (40)

Although the population size of other methods is a known factor, the number of solution quality evaluations must be calculated [38]. The best cost reported in Table 3 and the best power loss reported in Table 4 can reveal that the proposed method is more effective than other compared methods in finding higher quality solutions since its best cost and loss are less than those of others. In fact, the proposed method can reduce the cost from $0.0776 to $3.342 and the loss from 0.006 to 2.22 MW corresponding to the improvement level from 0.01% to 0.417% for the cost and from 0.21% to 43.88% for the loss. The mean and the maximum cost indicate that the proposed method and other ones have approximately equal stabilization because the deviation is not high. The mean and the maximum power loss of other methods have not been reported for comparison. In addition, the number of

Figure 13. The best location of PVS found by MCOA for the case of reducing total power losses.

5.2. Comparisons of Result from MCOA and Other Method for the IEEE 30-Bus Power System

Comparisons of total fuel cost and total power losses from MCOA and other methods are shown inTables 3 and 4, respectively. In addition to the best, mean, and worst values of cost and loss, three otherfactors including population size, the maximum number of iterations, and the number of solutionquality evaluations are also mentioned for performance comparisons. Among the factors, populationsize and the number of solution quality evaluations of the proposed method are calculated by usingthe following equations:

Npop = NoGr×NoCoy, (39)

NoSqe = NoIter× [(NoGr×NoCoy) + (NpGr× 1)]. (40)

Although the population size of other methods is a known factor, the number of solution qualityevaluations must be calculated [38]. The best cost reported in Table 3 and the best power loss reportedin Table 4 can reveal that the proposed method is more effective than other compared methods infinding higher quality solutions since its best cost and loss are less than those of others. In fact,the proposed method can reduce the cost from $0.0776 to $3.342 and the loss from 0.006 to 2.22 MWcorresponding to the improvement level from 0.01% to 0.417% for the cost and from 0.21% to 43.88%for the loss. The mean and the maximum cost indicate that the proposed method and other ones haveapproximately equal stabilization because the deviation is not high. The mean and the maximumpower loss of other methods have not been reported for comparison. In addition, the number of

Energies 2019, 12, 4310 23 of 36

solution quality evaluations between the proposed and other ones should be mentioned. NoSqe of theproposed method is 2000 while that of others is much higher, from 4500 to 25,000 excluding SSO andISSO with NoSqe = 690. However, the best cost and the best loss from SSO and ISSO are higher thanthose of the proposed method. Consequently, it can be concluded that the proposed method is moreeffective than other ones for the IEEE 30-bus transmission power network.

Table 3. Comparison of EGFC obtained by MCOA and other methods for the IEEE 30-bus transmissionpower network.

Method MinimumCost ($)

Mean Cost($)

MaximumCost ($) Npop NoIter NoSqe

DE [4] 801.23 801.282 801.622 - - -DE [5] 799.2891 - - 50 500 25,000

EADPSO [10] 800.2276 800.2625 800.3274 50 250 12,500GPUA-PSO [12] 800.53 - - - - -DQLF-EGA [15] 799.56 - - 60 200 12,000

EGA-IPL [16] 799.56 799.6497 799.688 60 76 4560EGA-BM [17] 800.0435 800.12 800.159 60 200 12,000

GSA [22] 798.6751 798.9131 799.0284 100 200 20,000ABCA [23] 800.6600 800.8715 801.8674 - - -GWOA [26] 799.5585 - - - - -

MELMM [28] 800.0781 - - - - -MCBOM [29] 799.0353 - - 50 500 25,000

MSM [32] 800.5099 - - - - -ACO [33] 802.148 - - - - -

EACO [33] 802.097 - - - - -MFO [34] 799.072 - - 40 500 20,000SCA [37] 800.102 - - NA 500 NA

MSCA [37] 799.31 - - NA 500 NASSO [38] 802.2580 - - 20 30 ≈690ISSO [38] 798.9936 - - 20 30 ≈690MCOA 798.916 800.184 803.314 16 100 2000

Table 4. Comparison of TAPL obtained by MCOA and other methods for the IEEE 30-bus transmissionpower network.

Method MinimumTAPL (MW)

Mean TAPL(MW)

MaximumTAPL (MW) Npop NoIter NoSqe

DE [4] 3.38 - - - - -IPSO [9] 5.0732 - - 100 200 20,000

DQLF-EGA [15] 3.2008 - - 60 200 12,000EGA-IPL [16] 3.244 - - 60 75 4500

ABCA [23] 3.1078 - - - - -MSM [32] 3.1005 - - - - -MFO [34] 2.853 - - 40 500 20,000SCA [37] 2.9425 - - - - -

MSCA [37] 2.9334 - - - - -SSO [38] 3.8239 - - 20 30 ≈690ISSO [38] 2.8678 - - 20 30 ≈690MCOA 2.847 3.194 4.941 16 100 2000

5.3. Result Comparisons for the IEEE 57-Bus Transmission Power Network

In the section, the proposed method is compared to OCOA and other methods based on resultsfrom minimizing fuel cost and power losses of the IEEE 57-bus transmission power system. OCOAand MCOA are executed by setting NoGr = 4, NoCoy = 4, and NoIter = 250. Comparisons of cost andloss for the system are shown in Tables 5 and 6, while fitness function values of 50 runs and the best

Energies 2019, 12, 4310 24 of 36

convergence characteristic of OCOA and MCOA are shown in Figures 14–17. Comparisons betweenOCOA and MCOA show that MCOA can find less cost and less power loss than OCOA by $360.38and 2.03 MW. These better results are corresponding to the improvement level of 0.86% and 17.3%.In addition, the mean and the maximum cost and loss of MCOA are also less than those of OCOA.For the network, MCOA and OCOA can reach SR with 96% and 74%, respectively. Therefore, MCOAstill gets better ability of handling constraints than OCOA. With respect to graphic comparisons shownin Figures 14–17 for 50 successful runs and the best convergence characteristic, MCOA also showssmaller fluctuations, high number of better solutions, and faster convergence speed. Therefore, theproposed MCOA method is really more robust than OCOA in finding the best solutions, reachinghigher success rate, having better stabilization of the search process, and converging to a solution faster.

The comparisons between the proposed MCOA and other ones emphasize that the proposedmethod is more effective than other ones since it can find less cost and less loss than all methods.As compared to the second best method and the worst method, the proposed MCOA can find lesscost by $6.63 and $450.81, and less power loss by 2.81 MW and 2.77 MW. The results were convertedto improvement level and this is 0.02% and 1.07% for cost, and 2.82% and 22.2% for power loss.Furthermore, the proposed method is also faster than nearly all methods because NoSqe of the proposedmethod is 5000 while that of other is from 5000 to 110,000 excluding SSO and ISSO with NoSqe = 1700.Therefore, it can be concluded that the proposed method is really effective for the IEEE 57-bustransmission power network.


Method MinimumCost ($)

Mean Cost($)

MaximumCost ($) Npop NoIter NoSqe

EADPSO [10] 41,697.54 - - 50 150 7500PSO [11] 42,109.723 44,688.420 4,9320.666 25 200 5000

PG-PSO [11] 41,688.500 42,032.706 4,4748.034 25 200 5000ITLBO [18] 41,638.382 - - - - -GSA [22] 41,695.871 - - - - -

ABCA [23] 41,693.958 - - 70 200 14,000IICM [24] 41,738.44 - - 220 500 110,000COA [25] 41,901.997 42,176.351 43,982.642 - - -

GBBICM [27] 41,715.710 - - - - -DSM [30] 41,686.82 - - - - -MSM [32] 41,673.723 - - - - -

BA[35] 41,716 41,808 41,889.0 - - -IBA [35] 41,673 41,720 41,790 - - -TSA [36] 41,685.07 41,687 41,689.05 10 1000 10,000SSO [38] 41,734.337 - - 30 50 ≈1700ISSO [38] 41,665.540 - - 30 50 ≈1700

OCOA 42,019.298 51,348.818 84,053.088 16 250 5000MCOA 41,658.913 42,158.042 44,411.362 16 250 5000

Energies 2019, 12, 4310 25 of 36

Table 6. Comparison of TAPL obtained by MCOA and other methods for the IEEE 57-bus transmissionpower network.

Method MinimumLoss (MW)

Mean Loss(MW)

MaximumLoss (MW) Npop NoIter NoSqe

IDE [8] 10.5581 - - - - -IICM [24] 11.8826 - - 220 500 110,000

SKHM [31] 10.6877 11.111 12.0016 60 200 12,000TSA [36] 12.473 12.54 12.60 10 1000 10,000SSO [38] 10.6144 - - 30 50 ≈1700ISSO [38] 9.9843 - - 30 50 ≈1700

OCOA 11.731 16.176 31.431 16 250 5000MCOA 9.703 19.285 39.713 16 250 5000


OCOA 42,019.298 51,348.818 84,053.088 16 250 5000 MCOA 41,658.913 42,158.042 44,411.362 16 250 5000

Table 6. Comparison of TAPL obtained by MCOA and other methods for the IEEE 57-bus transmission power network.

Method Minimum Loss

(MW) Mean Loss

(MW) Maximum Loss

(MW) Npop NoIter NoSqe

IDE [8] 10.5581 - - - - - IICM [24] 11.8826 - - 220 500 110,000

SKHM [31]

10.6877 11.111 12.0016 60 200 12,000

TSA [36] 12.473 12.54 12.60 10 1000 10,000 SSO [38] 10.6144 - - 30 50 ≈1700 ISSO [38] 9.9843 - - 30 50 ≈1700

OCOA 11.731 16.176 31.431 16 250 5000 MCOA 9.703 19.285 39.713 16 250 5000



Fitn

ess

func

tion

Figure 14. The fitness function values of 50 successful runs obtained by OCOA and MCOA for theEGFC objective of the IEEE 57-bus transmission power network.


OCOA 42,019.298 51,348.818 84,053.088 16 250 5000 MCOA 41,658.913 42,158.042 44,411.362 16 250 5000

Table 6. Comparison of TAPL obtained by MCOA and other methods for the IEEE 57-bus transmission power network.

Method Minimum Loss

(MW) Mean Loss

(MW) Maximum Loss

(MW) Npop NoIter NoSqe

IDE [8] 10.5581 - - - - - IICM [24] 11.8826 - - 220 500 110,000

SKHM [31]

10.6877 11.111 12.0016 60 200 12,000

TSA [36] 12.473 12.54 12.60 10 1000 10,000 SSO [38] 10.6144 - - 30 50 ≈1700 ISSO [38] 9.9843 - - 30 50 ≈1700

OCOA 11.731 16.176 31.431 16 250 5000 MCOA 9.703 19.285 39.713 16 250 5000



Fitn

ess

func

tion

Figure 15. Fitness function iteration characteristic of the best run from OCOA and MCOA for the EGFCobjective of the IEEE 57-bus transmission power network.

Energies 2019, 12, 4310 26 of 36Energies 2019, 12, x FOR PEER REVIEW 26 of 36


Figure 17. Fitness function iteration characteristic of the best run from OCOA and MCOA for the TAPL objective of the IEEE 57-bus transmission power network.


In the section, the IEEE 118-bus transmission power network is employed as the most complicated study case with 118 buses and 130 control variables. NoGr, NoCoy and NoIter were, respectively, set to 5, 5, and 300. From these settings, Npop and NoSqe from OCOA and MCOA are calculated to be 25 and 9000. Result comparisons are shown in Table 7. For the test system, both MCOA and OCOA cannot reach SR of 100% due to the impact of the large scale system. However, the success rate of the proposed MCOA is much higher than that of OCOA since SR of OCOA and MCOA is, respectively, 16% and 52%. SR from other compared methods in Table 7 was not reported in previous studies, so we cannot compare the constraint handling ability of the proposed method and these ones. For comparison of the best cost, the proposed method can find less cost than OCOA by $2469.78 corresponding to the improvement of 1.87%. As observing minimum cost of all methods, the proposed method can reach a better optimal solution than all methods with less cost by $168.91 to $9098.85, corresponding to the improvement from 0.13% to 6.55%. The reported values are not compared to GPUA-PSO because study [12] did not report control variables for checking. The mean

Fitn

ess

func

tion

Fitn

ess

func

tion

Figure 16. The fitness function values of 50 successful runs obtained by OCOA and MCOA for theTAPL objective of the IEEE 57-bus transmission power network.



Figure 17. Fitness function iteration characteristic of the best run from OCOA and MCOA for the TAPL objective of the IEEE 57-bus transmission power network.


In the section, the IEEE 118-bus transmission power network is employed as the most complicated study case with 118 buses and 130 control variables. NoGr, NoCoy and NoIter were, respectively, set to 5, 5, and 300. From these settings, Npop and NoSqe from OCOA and MCOA are calculated to be 25 and 9000. Result comparisons are shown in Table 7. For the test system, both MCOA and OCOA cannot reach SR of 100% due to the impact of the large scale system. However, the success rate of the proposed MCOA is much higher than that of OCOA since SR of OCOA and MCOA is, respectively, 16% and 52%. SR from other compared methods in Table 7 was not reported in previous studies, so we cannot compare the constraint handling ability of the proposed method and these ones. For comparison of the best cost, the proposed method can find less cost than OCOA by $2469.78 corresponding to the improvement of 1.87%. As observing minimum cost of all methods, the proposed method can reach a better optimal solution than all methods with less cost by $168.91 to $9098.85, corresponding to the improvement from 0.13% to 6.55%. The reported values are not compared to GPUA-PSO because study [12] did not report control variables for checking. The mean

Fitn

ess

func

tion

Fitn

ess

func

tion

Figure 17. Fitness function iteration characteristic of the best run from OCOA and MCOA for the TAPLobjective of the IEEE 57-bus transmission power network.


In the section, the IEEE 118-bus transmission power network is employed as the most complicatedstudy case with 118 buses and 130 control variables. NoGr, NoCoy and NoIter were, respectively, set to5, 5, and 300. From these settings, Npop and NoSqe from OCOA and MCOA are calculated to be 25 and9000. Result comparisons are shown in Table 7. For the test system, both MCOA and OCOA cannotreach SR of 100% due to the impact of the large scale system. However, the success rate of the proposedMCOA is much higher than that of OCOA since SR of OCOA and MCOA is, respectively, 16% and52%. SR from other compared methods in Table 7 was not reported in previous studies, so we cannotcompare the constraint handling ability of the proposed method and these ones. For comparison ofthe best cost, the proposed method can find less cost than OCOA by $2469.78 corresponding to theimprovement of 1.87%. As observing minimum cost of all methods, the proposed method can reach abetter optimal solution than all methods with less cost by $168.91 to $9098.85, corresponding to theimprovement from 0.13% to 6.55%. The reported values are not compared to GPUA-PSO becausestudy [12] did not report control variables for checking. The mean cost and the worst cost of theproposed method are less than those of all methods. Furthermore, NoSqe of the proposed method is

Energies 2019, 12, 4310 27 of 36

9000 while that of others is from 9000 to 225,000. Therefore, the proposed method is superior to allcompared methods and it is really powerful for the IEEE 118-bus transmission power network.


Method MinimumEGFC ($)

Mean EGFC($)

MaximumEGFC ($) Npop NoIter NoSqe

PSO [11] 145,520.0109 158,596.1725 184,686.824 40 250 10,000PG-PSO [11] 139,604.1326 152,204.2608 170,022.972 40 250 10,000

GPUA-PSO [12] 129,627.03* - - - 500 -MPM [21] 130,114.429 - - - - -COA [25] 133,110.4316 138,260.4028 153,110.431 - - -

MCBOM [29] 135,121.570 - - 90 2500 225,000ACO [33] 138,809.3896 142,189.2573 149,097.287 - 100 -

EACO [33] 138,757.7521 142,062.4122 148,567.872 - 100 -SSO [38] 132,080.4118 - - 40 200 ≈9000ISSO [38] 129,879.45361 - - 40 200 ≈9000

OCOA 132,180.3208 135,442.69 139,587.82 25 300 9000MCOA 129,710.5410 132,861.75 134,172.02 25 300 9000

Note: Method with * did not report optimal solution.

In the Appendix A, optimal control variables reached by the proposed MCOA are presentedin Tables A1–A4.

5.5. Discussion

OCOA and MCOA are meta-heuristic algorithms based on randomizations similarly to othermeta-heuristic algorithms such as GA, PSO, DE, etc. In the structure of search process, OCOA isparticularly dependent on randomizations, especially the second generation in Equation (17) withthree variable selection choices, varn,pr1,g, varn,pr2,g, and varn,rd. Furthermore, the selection of one outof the three variables is decided by another randomization condition. That is either λ4 < 1/NoCVor λ4 > 1/NoCV + 0.5. In addition, in the step of exchanging solutions, OCOA is also based on onemore randomization condition of λ5 < 0.005.NoCoy2. The randomizations are not effective, but theyare time consuming. Therefore, we modified Equation (17) and canceled using the condition forexchanging solutions. The task can reduce ineffective computation steps as well as shorten executiontime. As a result, MCOA can avoid the ineffective randomizations. However, MCOA cannot eliminaterandomizations completely as shown in Equation (21) and (22) and the task of exchanging solutions.The randomization in Equations (21) and (22) are the characteristic of meta-heuristic algorithmsbecause the randomizations can support production of different solutions similarly to PSO, DE, etc.The randomization in exchanging solutions is also useful for avoiding lumping all solutions in eachgroup, leading to a very low possibility of converging local optimum. Consequently, MCOA solvedthe OPF problem more effectively than OCOA. In Sections 5.1.1 and 5.1.2, we ran OCOA and MCOAby setting different values to the number of groups and the number of coyotes in each group. Theresult comparisons indicated that MCOA was completely more effective than OCOA. For each studycase, we ran the two methods, and 50 trials and results in terms of minimum, mean, and maximumfitness are reported in Tables 1 and 2 and Figures 6–11. The result comparisons indicate that MCOAcan reach better solutions than OCOA with less fuel cost and less power losses.

For solving the three systems, the proposed MCOA tackled several disadvantages that its originalmethod suffered such as slowly finding high quality solutions, having a high impact on randomizations,owning a high number of computation steps, and using a long simulation time. As shown in sectionsabove, the proposed method could reach better results than OCOA such as

Energies 2019, 12, 4310 28 of 36

1 Finding higher quality solutions: the best fuel cost and the best total power loss of MCOA areless than those from OCOA for all study cases,

2 Having more stable search ability: figures showing fitness function of 50 trial runs indicated thatapproximately all runs of MCOA had lower fitness function than those of OCOA,

3 Reducing simulation time: MCOA can reduce a number of computation steps by cancelingrandomizations in the first and the second generations. Furthermore, the fast search speed canreduce the number of iterations and shorten simulation time.

However, we had to cope with several difficulties to reach high performance for the search processof the proposed MCOA in dealing with the OPF problem, especially for a large scale system with 118buses and a high number of control variable cases such as the case with installation of PVS for the IEEE30-bus system. As we knew, the proposed MCOA method has three basic control parameters includingthe number of coyotes in each group NoCoy, the number of groups NoGr, and the number of iterationsNoIter. The population size of the proposed MCOA method is equal to (NoGr × NoCoy) meanwhile thenumber of new solutions produced for each iteration and for one run is, respectively, (NoGr × NoCoy +

NoGr) and (NoGr × NoCoy + NoGr) × NoIter. Therefore, the task of tuning the most appropriate valuesfor NoCoy, NoGr, and NoIter is not easy. In fact, during the implementation of the proposed MCOA weconsidered the comparison criterion, which is the number of solution quality evaluations NoSqe shownin Equation (40). As shown in Tables 1 and 2, we set different values for NoCoy and NoGr but fixedNoIter = 100. However, it is just one setting of the trial settings for reaching the best results of MCOA.During the implementation, we also reduced NoIter and increased the result of (NoGr × NoCoy). Forthe setting of (NoGr × NoCoy), we also increased NoGr or NoCoy and decreased the remaining one. As aresult, we concluded the most appropriate setting for the parameters and we applied the settings forall study cases of the considered system. Results shown in sections above indicated that four was thebest value for both NoGr and NoCoy for the IEEE 30-bus system and IEEE 57-bus system but five wasthe best value for both NoGr and NoCoy for the IEEE 118-bus system. It is clear that the setting of thecontrol parameters of MCOA is not an easy task and it must take long simulation time. The difficultywill become more serious if the selection of control variables is not certain and there are many choicesof control variables. Fortunately, the OPF problem is complicated but the selection of control variablesis predetermined by using the MatPower program. As using the Matpower program, input data mustbe active power output of all generators excluding generator at slack bus, voltage of all generators,reactive power output of capacitors, and tap changer of transformers. Therefore, all the variables arepredetermined as control variables and there is no difficulty for the selection of control variables. Forthe case of installing PVS, the position of one load bus must be selected for providing 2.0 MW to theload and reducing 2.0 MW supplied from the transmission lines. Therefore, the selection of controlvariables was also an easy task for the more complicated case of installing PVS. In the future, we willconsider other clean power sources such as wind turbines and hydropower plants as a real powersystem. In addition, a multi-period problem will be studied instead of single period problem in thisstudy. In daytime hours, PVS together with other sources will supply electricity to the load but thepresence of PVS will not be taken into account at night. The task of the new problem is to determinethe best location for installing PVS and wind turbines meanwhile the location of hydropower plantsand thermal power plants is known.

6. Conclusions

In this paper, a novel coyote optimization algorithm was proposed for solving the OPF problem.The proposed method was developed by modifying two techniques of generating new solutions andone technique of exchanging solutions among available groups. By testing on IEEE 30, 57, and 118-bustransmission power networks with two cases of minimizing electric generation fuel cost and total activepower losses, the proposed method was demonstrated to be more effective and robust than OCOA interms of better quality of the best solution, better stabilization of search process, faster convergence,and higher success rate. The result comparisons are as follows:

Energies 2019, 12, 4310 29 of 36

1 The best success rate of the proposed method the three systems was, respectively, 100%, 96%,and 52%, meanwhile that of OCOA was, respectively, 88%, 74%, and 16%.

2 The improvement level of cost from MCOA over OCOA could be up to 3.54% while theimprovement level for power loss could be up to 30.21%.

As compared to other methods, the method could be more effective in finding better solutionswith a faster manner. The result comparisons can be summarized as follows:

1. The proposed method could obtain improvement levels of cost and loss as 0.417% and 43.88%for the IEEE 30-bus transmission power network, and 1.07% and 22.2% for the IEEE 57-bustransmission power network, while the improvement level of cost for IEEE 118-bus transmissionpower network was 6.55%.

2. The proposed method used either an equal or smaller number of solution quality evaluations.

For further investigation of the performance, OCOA and MCOA were run to install a PVS in theIEEE 30-bus system. The results were promising as follows:

1 Both OCOA and MCOA could reduce the cost by 0.7%–8.5% and the power loss by 8.5% and3.8%.

2 As compared to OCOA, MCOA could reduce fuel cost and power loss by 0.5% and24.36%, respectively.

Consequently, it is recommended that MCOA is a favorable optimization tool for solving the OPFproblem and for more complicated studies with the presence of renewable energies.

Author Contributions: Z.L. and Y.C. simulated results and wrote important parts of the article. L.V.D. and X.Y.selected references, formatted the whole paper, summarized results in Tables, and wrote the introduction. T.T.N.edited the whole paper.

Funding: This research was funded by the National Natural Science Foundation of China (NSFC) under projectNo. 51520105011.

Conflicts of Interest: The authors declare that there is no conflict of interests regarding the publication of this paper.

Abbreviations

ABCA Artificial bee colony algorithmACO Ant colony optimizationAGSO Adaptive group search optimizationBA Bat AlgorithmBSA Backtracking Search algorithmCOA Cuckoo optimization algorithmDSM Differential search methodEACO Enhanced ant colony optimizationFA Firefly algorithmGBBICM Gaussian bare-bones imperialist competitive methodGSA Gravitational search algorithmGWOA Grey Wolf Optimization algorithmIBA Improved bat algorithmIEEE Institute of Electrical and Electronics EngineersIHBMO Improved honey bee mating optimizationIICM Improved imperialist competitive methodISSO Improved social spider optimizationITLBO Improved teaching learning based optimizationMCBOM Modified Colliding Bodies Optimization methodMELMM Modified electromagnetism-like Mechanism methodMFO Moth-flame Optimizer

Energies 2019, 12, 4310 30 of 36

SaDE Self-adaptive Differential EvolutionSCA Sine-Cosine algorithmSKHM Stud krill herd methodSOSA Symbiotic Organism Search algorithmSSO Social spider optimizationTSA Tree-seed algorithmMPM Mathematical programming methodMSCA Modified Sine-Cosine algorithmMSM Moth swarm methodNomenclatureB ji, G ji Susceptance and conductance of the feeder connecting bus j and bus iC1, C2, C3, C4 Penalty factorsCoycen,g The center solution of the gth groupCoym,g The mth solution in the gth groupCoypr1,g, Coypr2,g Two randomly picked solutions from the gth groupCoyworst,g The worst solution in the gth groupCoyGbest The best solution in the whole populationCVari The control variable set of the ith solution

∆MVAbr,iThe penalty term for the violation of apparent power of the brth feeder correspondingto the ith solution

∆P1,iThe penalty term for the violation of active power at slack bus corresponding to theith solution

∆Qk,iThe penalty term for the violation of reactive power at the kth generator buscorresponding to the ith solution

∆VLl,iThe penalty term for the violation of voltage at the lth load bus corresponding to theith solution

EGFC The total electric generation fuel cost of all thermal power plants

EGFCk,iThe electric generation fuel cost of the kth thermal power plant corresponding to theith solution

FTi Fitness function of the ith solutionf j, fi Phase angles of voltage at the jth bus and the ith busFTworst,g Fitness value of the worst solution in the gth groupIter Current iterationλ1, λ2, λ3, λ4, λ5, λ6,λ7, λ8, λ9, λ10, λ11

Randomly generated numbers in the range from 0 to 1

MVAbr,i The apparent power flowing through the brth transmission of the ith solutionMVAmax

br The highest apparent power of the brth feederNoB The number of buses in the considered transmission power networkNoBR The number of transmission branches in the considered transmission gridNoCV The number of control variables included in each solutionNoCoy The number of solutions in each groupNoGr The number of groupsNoIter The maximum number of iterationsNoL The number of load buses in the considered transmission gridNoSqe The number of solution quality evaluationsNoS The number of solutions or population sizeNoSVC The number of static VAR compensators in the considered transmission gridNoT The number of transformersNpop Population sizeP1,i Active power of the thermal power plant at slack bus corresponding to the ith solutionPj, Qj Active and reactive power generated at the jth bus

Pk,i, Qk,i, Vk,iActive power, reactive power and voltage of the kth thermal power plantcorresponding to the ith solution

Energies 2019, 12, 4310 31 of 36

Pmink , Pmax

k The smallest and highest active power output of the kth thermal power plantPmin

1 , Pmax1 The smallest and highest active power output of the thermal power plant at slack bus

PLj Active power demand of the load at the jth busPLk The active power demand at the kth busQmin

k , Qmaxk The smallest and highest reactive power output of the kth thermal power plant

SP Site of photovoltaic systemSPi Site of photovoltaic system corresponding to the ith solution

SVCc,iThe reactive power of the cth static VAR compensator corresponding to the ithsolution

SVCj The reactive power of static VAR compensator generated at the jth busSVCmin

c , SVCmaxc Lower bound and upper bound of the cth static VAR compensator

Tt,i Tap changer of the tth transformer corresponding to the ith solutionTmin, Tmax The smallest tap changer and the highest tap changer of transformersVmin, Vmax Lower bound and upper bound of voltage of thermal power plant generatorsVi, Vj Magnitude of voltage at the ith bus and the jth busVarn,g The nth control variable of the solution in the gth groupVLl,i Voltage of load at the lth load bus corresponding to the ith solutionVLmin, VLmax Lower bound and upper bound of load bus voltage

Appendix A

Table A1. Control variables obtained by the proposed MCOA method for the IEEE 30-bus transmissionpower network.

Variable EGFC Objective TAPL Objective

PG1 177.2642 51.2489PG2 48.7616 80.0000PG5 21.1802 50.0000PG8 20.6942 35.0000PG11 12.0994 30.0000PG13 12.0066 39.9980VG1 1.1000 1.1000VG2 1.0879 1.1000VG5 1.0608 1.0820VG8 1.0682 1.0899VG11 1.0999 1.1000VG13 1.1000 1.1000Qc1 5.0000 4.3753Qc2 4.7782 0.0001Qc3 4.3765 4.9727Qc4 4.5808 5.0000Qc5 4.8757 5.0000Qc6 5.0000 5.0000Qc7 3.3788 1.0161Qc8 4.9352 5.0000Qc9 2.7671 1.2002T11 1.0389 1.0598T12 0.9000 0.9066T15 0.9827 0.9758T36 0.9658 0.9667

Energies 2019, 12, 4310 32 of 36

Table A2. Optimal solutions obtained by MCOA for the case with PVS in the IEEE 30-bus system.

Variable Optimize Total Power Losses Optimize Total Fuel Cost

PG1 50.1323 177.3232PG2 79.0242 47.7038PG5 50 21.0492PG8 34.9821 21.3667PG11 30 10.5512PG13 40 12.0120VG1 1.1 1.1000VG2 1.096 1.0862VG5 1.0806 1.0533VG8 1.0888 1.0649VG11 1.1 1.1000VG13 1.1 1.0984Qc1 0 4.9479Qc2 4.7165 4.8779Qc3 5 4.8243Qc4 5 3.9381Qc5 5 2.6789Qc6 4.2267 4.6555Qc7 1.2101 3.0685Qc8 4.9853 3.9158Qc9 3.5789 4.9469T11 0.9434 0.9508T12 1.1 0.9977T15 1.0115 0.9883T36 1.0037 0.9762SP 30 15


Variable Fuel Cost Minimization Power Loss Minimization

PG1 144.9455 209.1483PG2 89.9556 3.0094PG3 44.8814 140PG6 64.8987 100PG8 460.005 299.3124PG9 99.3167 99.2739PG12 361.3249 409.7594VG1 1.0981 1.1VG2 1.091 1.0944VG3 1.0907 1.0999VG6 1.1 1.0985VG8 1.1 1.0969VG9 1.0805 1.0793VG12 1.0938 1.0893Qc1 9.8097 0Qc2 5.1456 5.0375Qc3 6.2944 6.3T1 1.1 1.1T2 1.0981 0.9T3 1.0701 1.005T4 1.0309 0.9001T5 0.9416 0.95T6 1.0057 0.9642T7 0.9885 0.9866T8 0.9467 0.9005

Energies 2019, 12, 4310 33 of 36

Table A3. Cont.

Variable Fuel Cost Minimization Power Loss Minimization

T9 0.9 0.9002T10 0.987 0.9899T11 0.953 0.9945T12 0.9808 1.0009T13 0.9518 0.9524T14 0.9906 1.0826T15 1.0194 1.0722T16 0.9794 0.9279T17 1.0053 0.9868


ControlVariable Value Control

Variable Value ControlVariable Value

P1 (MW) 24.8557 P103 (MW) 38.0638 V76 (PU) 1.004

P4 (MW) 0.023 P104 (MW) 0.4972 V77 (PU) 1.0311

P6 (MW) 0.0015 P105 (MW) 1.5594 V80 (PU) 1.0435

P8 (MW) 0.0767 P107 (MW) 34.5069 V85 (PU) 1.028

P10 (MW) 399.7336 P110 (MW) 11.6535 V87 (PU) 1.0372

P12 (MW) 86.179 P111 (MW) 36.2049 V89 (PU) 1.0447

P15 (MW) 20.4286 P112 (MW) 33.962 V90 (PU) 1.0211

P18 (MW) 11.7113 P113 (MW) 0.3254 V91 (PU) 1.0209

P19 (MW) 22.2119 P116 (MW) 0.0002 V92 (PU) 1.0304

P24 (MW) 0.236 V1 (PU) 1.0005 V99 (PU) 1.0332

P25 (MW) 194.5895 V4 (PU) 1.0296 V100 (PU) 1.0379

P26 (MW) 281.4839 V6 (PU) 1.0211 V103 (PU) 1.0335

P27 (MW) 11.4272 V8 (PU) 1.0781 V104 (PU) 1.0188

P31 (MW) 7.3547 V10 (PU) 1.0929 V105 (PU) 1.0169

P32 (MW) 14.089 V12 (PU) 1.0157 V107 (PU) 1.0118

P34 (MW) 3.3261 V15 (PU) 1.0162 V110 (PU) 1.0527

P36 (MW) 4.4716 V18 (PU) 1.0202 V111 (PU) 1.0867

P40 (MW) 46.8342 V19 (PU) 1.0167 V112 (PU) 1.0474

P42 (MW) 38.953 V24 (PU) 1.0394 V113 (PU) 1.0263

P46 (MW) 19.2667 V25 (PU) 1.0587 V116 (PU) 1.0505

P49 (MW) 193.4075 V26 (PU) 1.0871 SVC5 (MVAr) −0.1794

P54 (MW) 49.3797 V27 (PU) 1.0211 SVC34 (MVAr) 0.0217

P55 (MW) 32.1311 V31 (PU) 1.0125 SVC37 (MVAr) −0.0246

P56 (MW) 37.4034 V32 (PU) 1.0187 SVC44 (MVAr) 0.0024

P59 (MW) 149.0935 V34 (PU) 1.0257 SVC45 (MVAr) 0.0011

P61 (MW) 147.9593 V36 (PU) 1.0232 SVC46 (MVAr) 0.0146

P62 (MW) 0.009 V40 (PU) 1.0122 SVC48 (MVAr) 0.0047

Energies 2019, 12, 4310 34 of 36

Table A4. Cont.

ControlVariable Value Control

Variable Value ControlVariable Value

P65 (MW) 352.0099 V42 (PU) 1.0119 SVC74 (MVAr) 0.0026

P66 (MW) 349.0427 V46 (PU) 1.032 SVC79 (MVAr) 0.0593

P70 (MW) 0.1006 V49 (PU) 1.043 SVC82 (MVAr) 0.0059

P72 (MW) 0.0192 V54 (PU) 1.0165 SVC83 (MVAr) 0.1876

P73 (MW) 0.7076 V55 (PU) 1.0151 SVC105 (MVAr) 0.0983

P74 (MW) 14.0153 V56 (PU) 1.0152 SVC107 (MVAr) 0

P76 (MW) 35.1036 V59 (PU) 1.0329 SVC110 (MVAr) 0.0069

P77 (MW) 0.0044 V61 (PU) 1.0362 T8 (pu) 1.0361

P80 (MW) 428.8456 V62 (PU) 1.0363 T32 (pu) 1.0135

P85 (MW) 0.0897 V65 (PU) 1.0626 T36 (pu) 1.0298

P87 (MW) 3.6106 V66 (PU) 1.0535 T51 (pu) 1.0152

P89 (MW) 498.136 V69 (PU) 1.066 T93 (pu) 0.9999

P90 (MW) 0.0061 V70 (PU) 1.0398 T95 (pu) 1.0153

P91 (MW) 0.0414 V72 (PU) 1.0396 T102 (pu) 0.9929

P92 (MW) 0.0123 V73 (PU) 1.0446 T107 (pu) 0.9344

P99 (MW) 0.3744 V74 (PU) 1.0135 T127 (pu) 0.9971

P100 (MW) 229.5969 - - - -

References

1. Nguyen, T.T.; Quynh, N.V.; Van Dai, L. Improved Firefly Algorithm: A Novel Method for Optimal Operationof Thermal Generating Units. Complexity 2018, 2018. [CrossRef]

2. Nguyen, T.; Vo, D.; Vu Quynh, N.; Van Dai, L. Modified cuckoo search algorithm: A novel method tominimize the fuel cost. Energies 2018, 11, 1328. [CrossRef]

3. Ongsakul, W.; Tantimaporn, T. Optimal power flow by improved evolutionary programming. Electr. PowerCompon. Syst. 2006, 34, 79–95. [CrossRef]

4. El-Fergany, A.A.; Hasanien, H.M. Single and multi-objective optimal power flow using grey wolf optimizerand differential evolution algorithms. Electr. Power Compon. Syst. 2015, 43, 1548–1559. [CrossRef]

5. El Ela, A.A.; Abido, M.A.; Spea, S.R. Optimal power flow using differential evolution algorithm. Electr. PowerSyst. Res. 2010, 80, 878–885. [CrossRef]

6. Thitithamrongchai, C.; Eua-Arporn, B. Self-adaptive differential evolution based optimal power flow forunits with non-smooth fuel cost functions. J. Electr. Syst. 2007, 3, 88–99.

7. Sayah, S.; Zehar, K. Modified differential evolution algorithm for optimal power flow with non-smooth costfunctions. Energy Convers. Manag. 2008, 49, 3036–3042. [CrossRef]

8. Shaheen, A.M.; El-Sehiemy, R.A.; Farrag, S.M. Solving multi-objective optimal power flow problem viaforced initialised differential evolution algorithm. IET Gener. Transm. Distrib. 2016, 10, 1634–1647. [CrossRef]

9. Niknam, T.; Narimani, M.R.; Aghaei, J.; Azizipanah-Abarghooee, R. Improved particle swarm optimisationfor multi-objective optimal power flow considering the cost, loss, emission and voltage stability index. IETGener. Transm. Distrib. 2012, 6, 515–527. [CrossRef]

10. Vaisakh, K.; Srinivas, L.R.; Meah, K. Genetic evolving ant direction particle swarm optimization algorithmfor optimal power flow with non-smooth cost functions and statistical analysis. Appl. Soft Comput. 2013, 13,4579–4593. [CrossRef]

11. Vo, D.N.; Schegner, P. An Improved Particle Swarm Optimization for Optimal Power Flow. In Meta-HeuristicsOptimization Algorithms in Engineering, Business, Economics, and Finance; IGI Global: Hershey, PA, USA, 2013;pp. 1–40.

http://dx.doi.org/10.1155/2018/7267593

http://dx.doi.org/10.3390/en11061328

http://dx.doi.org/10.1080/15325000691001458

http://dx.doi.org/10.1080/15325008.2015.1041625

http://dx.doi.org/10.1016/j.epsr.2009.12.018

http://dx.doi.org/10.1016/j.enconman.2008.06.014

http://dx.doi.org/10.1049/iet-gtd.2015.0892


http://dx.doi.org/10.1016/j.asoc.2013.07.002

Energies 2019, 12, 4310 35 of 36

12. Roberge, V.; Tarbouchi, M.; Okou, F. Optimal power flow based on parallel metaheuristics for graphicsprocessing units. Electr. Power Syst. Res. 2016, 140, 344–353. [CrossRef]

13. Marcelino, C.G.; Almeida, P.E.; Wanner, E.F.; Baumann, M.; Weil, M.; Carvalho, L.M.; Miranda, V. Solvingsecurity constrained optimal power flow problems: A hybrid evolutionary approach. Appl. Intell. 2018, 48,3672–3690. [CrossRef]

14. Saini, A.; Chaturvedi, D.K.; Saxena, A.K. Optimal power flow solution: a GA-fuzzy system approach. Int. J.Emerg. Electr. Power Syst. 2006, 5. [CrossRef]

15. Kumari, M.S.; Maheswarapu, S. Enhanced genetic algorithm based computation technique for multi-objectiveoptimal power flow solution. Int. J. Electr. Power Energy Syst. 2010, 32, 736–742. [CrossRef]

16. Reddy, S.S.; Bijwe, P.R.; Abhyankar, A.R. Faster evolutionary algorithm based optimal power flow usingincremental variables. Int. J. Electr. Power Energy Syst. 2014, 54, 198–210. [CrossRef]

17. Reddy, S.S.; Bijwe, P.R. Efficiency improvements in meta-heuristic algorithms to solve the optimal powerflow problem. Int. J. Electr. Power Energy Syst. 2016, 82, 288–302. [CrossRef]

18. Shabanpour-Haghighi, A.; Seifi, A.R.; Niknam, T. A modified teaching–learning based optimization formulti-objective optimal power flow problem. Energy Convers. Manag. 2014, 77, 597–607. [CrossRef]

19. Daryani, N.; Hagh, M.T.; Teimourzadeh, S. Adaptive group search optimization algorithm for multi-objectiveoptimal power flow problem. Appl. Soft Comput. 2016, 38, 1012–1024. [CrossRef]

20. Niknam, T.; Narimani, M.R.; Aghaei, J.; Tabatabaei, S.; Nayeripour, M. Modified honey bee matingoptimisation to solve dynamic optimal power flow considering generator constraints. IET Gener. Transm.Distrib. 2011, 5, 989–1002. [CrossRef]

21. Ara, A.L.; Kazemi, A.; Gahramani, S.; Behshad, M. Optimal reactive power flow using multi-objectivemathematical programming. Sci. Iran. 2012, 19, 1829–1836. [CrossRef]

22. Duman, S.; Güvenç, U.; Sönmez, Y.; Yörükeren, N. Optimal power flow using gravitational search algorithm.Energy Convers. Manag. 2012, 59, 86–95. [CrossRef]

23. Adaryani, M.R.; Karami, A. Artificial bee colony algorithm for solving multi-objective optimal power flowproblem. Int. J. Electr. Power Energy Syst. 2013, 53, 219–230. [CrossRef]

24. Ghasemi, M.; Ghavidel, S.; Ghanbarian, M.M.; Gharibzadeh, M.; Vahed, A.A. Multi-objective optimal powerflow considering the cost, emission, voltage deviation and power losses using multi-objective modifiedimperialist competitive algorithm. Energy 2014, 78, 276–289. [CrossRef]

25. Le Anh, T.N.; Vo, D.N.; Ongsakul, W.; Vasant, P.; Ganesan, T. Cuckoo optimization algorithm for optimalpower flow. In Adaptation, Learning and Optimization, Proceedings of the 18th Asia Pacific Symposium on Intelligentand Evolutionary Systems; Springer: Berlin, Germany, 2015; Volume 1, pp. 479–493.

26. Ladumor, D.P.; Trivedi, I.N.; Bhesdadiya, R.H.; Jangir, P. A grey wolf optimizer algorithm for VoltageStability Enhancement. In Proceedings of the 2017 Third International Conference on Advances in Electrical,Electronics, Information, Communication and Bio-Informatics (AEEICB), Chennai, India, 27–28 February 2017.

27. Ghasemi, M.; Ghavidel, S.; Ghanbarian, M.M.; Gitizadeh, M. Multi-objective optimal electric power planningin the power system using Gaussian bare-bones imperialist competitive algorithm. Inf. Sci. 2015, 294,286–304. [CrossRef]

28. El-Hana Bouchekara, H.R.; Abido, M.A.; Chaib, A.E. Optimal power flow using an improvedelectromagnetism-like mechanism method. Electr. Power Compon. Syst. 2016, 44, 434–449. [CrossRef]

29. Bouchekara, H.R.E.H.; Chaib, A.E.; Abido, M.A.; El-Sehiemy, R.A. Optimal power flow using an ImprovedColliding Bodies Optimization algorithm. Appl. Soft Comput. 2016, 42, 119–131. [CrossRef]

30. Abaci, K.; Yamacli, V. Differential search algorithm for solving multi-objective optimal power flow problem.Int. J. Electr. Power Energy Syst. 2016, 1, 10–79. [CrossRef]

31. Pulluri, H.; Naresh, R.; Sharma, V. A solution network based on stud krill herd algorithm for optimal powerflow problems. Soft Comput. 2018, 22, 159–176. [CrossRef]

32. Mohamed, A.A.A.; Mohamed, Y.S.; El-Gaafary, A.A.; Hemeida, A.M. Optimal power flow using moth swarmalgorithm. Electr. Power Syst. Res. 2017, 142, 190–206. [CrossRef]

33. Raviprabakaran, V.; Subramanian, R.C. Enhanced ant colony optimization to solve the optimal power flowwith ecological emission. Int. J. Syst. Assur. Eng. Manag. 2018, 9, 58–65. [CrossRef]

34. Trivedi, I.N.; Jangir, P.; Parmar, S.A.; Jangir, N. Optimal power flow with voltage stability improvement andloss reduction in power system using Moth-Flame Optimizer. Neural Comput. Appl. 2018, 30, 1889–1904.[CrossRef]


http://dx.doi.org/10.1007/s10489-018-1167-5

http://dx.doi.org/10.2202/1553-779X.1091

http://dx.doi.org/10.1016/j.ijepes.2010.01.010






http://dx.doi.org/10.1016/j.scient.2012.07.010



http://dx.doi.org/10.1016/j.energy.2014.10.007

http://dx.doi.org/10.1016/j.ins.2014.09.051

http://dx.doi.org/10.1080/15325008.2015.1115919



http://dx.doi.org/10.1007/s00500-016-2319-3


http://dx.doi.org/10.1007/s13198-016-0471-x

http://dx.doi.org/10.1007/s00521-016-2794-6

Energies 2019, 12, 4310 36 of 36

35. Yuan, Y.; Wu, X.; Wang, P.; Yuan, X. Application of improved bat algorithm in optimal power flow problem.Appl. Intell. 2018, 48, 2304–2314. [CrossRef]

36. El-Fergany, A.A.; Hasanien, H.M. Tree-seed algorithm for solving optimal power flow problem in large-scalepower systems incorporating validations and comparisons. Appl. Soft Comput. 2018, 64, 307–316. [CrossRef]

37. Attia, A.F.; Sehiemy, R.A.E.; Hasanien, H.M. Optimal power flow solution in power systems using a novelSine-Cosine algorithm. Int. J. Electr. Power Energy Syst. 2018, 99, 331–343. [CrossRef]

38. Nguyen, T.T. A high performance social spider optimization algorithm for optimal power flow solution withsingle objective optimization. Energy 2019, 171, 218–240. [CrossRef]

39. Pierezan, J.; Coelho, L.D.S. Coyote Optimization Algorithm: A New Metaheuristic for Global OptimizationProblems. IEEE Congr. Evol. Comput. 2018, 1–8. [CrossRef]

40. Qais, M.H.; Hasanien, H.M.; Alghuwainem, S.; Nouh, A.S. Coyote optimization algorithm for parametersextraction of three-diode photovoltaic models of photovoltaic modules. Energy 2019, 187, 116001. [CrossRef]

41. Pierezan, J.; Maidl, G.; Yamao, E.M.; dos Santos Coelho, L.; Mariani, V.C. Cultural coyote optimizationalgorithm applied to a heavy duty gas turbine operation. Energy Convers. Manag. 2019, 199, 111932.[CrossRef]

42. Güvenç, U.; Kaymaz, E. Economic Dispatch Integrated Wind Power Using Coyote Optimization Algorithm.In Proceedings of the 2019 7th International Istanbul Smart Grids and Cities Congress and Fair (ICSG),Istanbul, Turkey, 25–26 April 2019; pp. 179–183.

43. Nguyen, T.T.; Vo, D.N. Improved social spider optimization algorithm for optimal reactive power dispatchproblem with different objectives. Neural Comput. Appl. 2019, 1–32. [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

http://dx.doi.org/10.1007/s10489-017-1081-2



http://dx.doi.org/10.1016/j.energy.2019.01.021

http://dx.doi.org/10.1109/CEC.2018.8477769

http://dx.doi.org/10.1016/j.energy.2019.116001

http://dx.doi.org/10.1016/j.enconman.2019.111932

http://dx.doi.org/10.1007/s00521-019-04073-4

http://creativecommons.org/

http://creativecommons.org/licenses/by/4.0/.

networks using a novel metaheuristic algorithm

Documents